The Twentieth Century 


Qfl 103 
. W55 
Copy 1 


Arithmetic. 


Part I. 


FOR 


PRIMARY SCHOOLS. 


BY 


ERNEST E. WEST, B.L., 

Graduate U. S- Naval Academy. 


Atlanta, Ga. : 

The Foote & Davies Company, 
1897. 



















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The Twentieth Century 


Arithmetic. 


Part I. 

FOR 

PRIMARY SCHOOLS. 


BY 

ERNEST E. WEST, B.L., 

Graduate U. 8. Naval Academy. 



Atlanta, Ga. : 

The Foote & Davies Company, 






Copyright, 1897 , 

By ERNEST E. WEST. 


PREFACE TO PART I. 


T HE idea of number arises when we limit quantity. The child and 
the savage have very imperfect ideas of number, because they 
are little concerned with estimating quantity. But as soon as 
they think of adapting means to an end, a more or less definite idea 
of measurement begins and thus the idea of number originates. It 
is evident, then, that to make the child’s idea of number definite and 
clear, the learning of number should begin with measuring some 
quantity. True ideas of number arise through the activity of the 
mind in dealing with objects in a constructive way. 

Just as words symbolize things, so numbers symbolize how many 
things; and the same method that is used in teaching new words 
may be used in teaching new numbers. A word is taught by 
(1) showing the object; (2) speaking the name ; (3) writing the name. 
A number may be taught by (1) counting the objects; (2) speaking 
the number; (3) writing the number. By persistly following this 
routine, the successive steps become familiar and proficiency is grad¬ 
ually attained. 

It is essential that the comparative values of the first ten numbers 
be fully appreciated ; but a complete analysis and appreciation of the 
value of large numbers is impossible. 

For learning these first ten numbers each pupil should have a hun¬ 
dred cubic centimeter blocks of wood and a rule one decimeter long 
divided into centimeters. The cost of these (about 25 cents) should 
not for a moment be thought of when compared to their real value as 
aides to the learner. The decimeter and cubic centimeter blocks are 
better than a foot and cubic inch blocks because, the relation of the 
former is the same as our scale of notation. The teacher should also 
have an abacus. This apparatus is certainly as much necessary to the 
teacher of primary arithmetic as proper apparatus is to the teacher of 
physics. 

When the pupil has learned the first ten numbers, the teacher may 
follow the steps as given in this volume with the assurance that a 
knowledge of our system of writing numbers and operations with 
them is being unfolded to the child in its proper order. 

ERNEST E. WEST. 


Atlanta, Ga., May 26, 1897. 



CONTENTS OF PART I. 


PAGE. 


Arabic System.2 

Numeration.5 

Review.9 

The Roman System ... 12 

Review. ,.15 

Diversions.16 

Reduction.16 

Addition.20 

Definitions.27 

When the sum of any col¬ 
umn is more than 9 . . 28 

Review.33 

Diversions ..34 

Multiplication.34 

Definitions.40 

To Multiply by 10, 100, 

etc.43 

To Multiply by any number 46 

Review.51 

Diversions.52 

Subtraction.53 

Definitions.55 

Review.58 


PAGE. 


Addition and Subtraction 59 

Diversions.61 

Division.62 

Definitions.63 

Short Division ..... 64 

Dong Division.68 

To Divide by 10, 100, etc. . 70 

Averages.72 

Review. 73 

Signs of Aggregation ... 76 

R e v i e w of Fundamental 

Operations.77 

Some Properties of Num¬ 
bers .79 

Casting Out Nines .... 80 

Diversions.82 

Tests of Divisibility ... 83 

Factors.85 

Greatest Common Divisor . 88 

G. G. D.—Second Method . 90 
Least Common Multiple . . 95 

L. 0. M.—Second Method . 98 
Review .loo 
























The Twentieth Century 
Arithmetic. 


Part I. 


/ 

// 

/// 

llll 

lllll 

i. 


Number. 

One. 

Two. 

Three. 

Four. 

Five. 


lllll / 
lllll // 
lllll III 
lllll llll 


Question. Number. 
How many? Six. 
How many? Seven. 
How many? Eight. 
How many? Nine. 


Question. 

How many? 

How many? 

How many? 

How many? 

How many? 

A number is a direct answer to the question “How 
many?” The science of numbers is called arithmetic. 

Arithmetic explains the simple properties of numbers and 
how to use them in making calculations. 


lllll lllll How many? Ten. 


People could not talk and could even think only a little if they did 
not have words ; words are simply aids to talking and thinking. So? 
without numbers, people could calculate only a very little; numbers 
are simply aids to calculating. 


2. NOTATION AND NUMERATION. 

It would be very inconvenient to express a number al¬ 
ways by dots or by marks; therefore certain characters 






2 


THE goth CENTURY ARITHMETIC. 


are used to express a few numbers and, combining these char¬ 
acters in certain ways, we express any number. 

This is just like expressing certain sounds with characters, 
■called letters, and combining these letters to form words. 

Numbers are expressed by characters in two systems 
the Arabic and the Roman. 

The Arabic system, because of its superiority, is used in 
making all calculations. The characters it employs are 
called figures. 

The Roman system is used chiefly in numbering chapters, 
time-pieces, kings, popes, and sometimes dates. The char¬ 
acters it employs are our ordinary capital letters. 

Notation is expressing a number with characters. 

Numeration is expressing a number with words. 


ARABIC SYSTEM. 


Scale. —A unit is one; as, one apple, one hundred, one x, 
one mark. Unus in Latin means one. 

1 Thousand. 1 Hundred. 1 Ten. 1 Unit. 



Hundred | 

| Hundred 

| Ten 

Ten 



Hundred | 

\ Hundred 

| | Ten 

| | Ten 

1 lllll 


Hundred | 

1 Hundred 

| | Ten 

| | Ten 

1 t 


Hundred j 

| Hundred 

| | Ten 

| | Ten 

1 lllll 


Hundred \ 

Hundred 

j | Ten 

| | Ten 



10 Hundreds. 

10 

Tens. 

10 Units. 


A group of partridges is 

called a covey, 

a group of sticks 

is 

called a bundle, a group 

of cows 

is called a herd; so in 


numbers, 

A group of 10 units is called a ten. 

A group of 10 tens is called a hundred. 

A group of 10 hundreds is called a thousand. 



































THE ARABIC SYSTEM. 


8 


We see from this that the scale of the Arabic system is 10, 
from which fact it is also called the decimal system. 

The Latin word decern means ten. 


4. Notation. —The figures used in this system are 

Figures. 0 123 4567 8 9 

Names. Naught One Two Three Four Five Six Seven Eight Nine 

Each figure represents one more than the preceding figure. 
Naming them in their order of increase is called counting 1 . 

5. All English words are formed by using only twenty-six 
letters; all Arabic numbers are formed by using only these ten 
figures. Numbers can be expressed by using more or fewer 
than ten figures; in fact, the system using twelve figures 
possesses some advantages over that using ten; but the deci¬ 
mal system is the only one used and consequently the only 
one that need be learned. Just as a word is written by 
placing its letters in their proper order, so a number is writ¬ 
ten by placing its figures in their proper order. 

If we change the order of the letters in cat to act, we 
change the word; so, also, if we change the order of figures 
in a number, we change the number. The orders are num¬ 
bered in succession from the right. 1492 (one four nine 
two) is a number in which 2 is in the 1st order, 9 in the 2d. 
What is in the 3d? the 4th? If we change the order of the 
figures to 2419, we make quite a different number. 


6. Write the following numbers in figures: 


Orders. 5th. 

4th. 

3d. 

2d. 

1st. 

1 . 

one 

seven 

seven 

six 

2 . 


three 

six 

five 

3 . 

five 

two 

eight 

naught 



4 


THE 20th CENTURY ARITHMETIC . 


Orders. 

5th. 

4th. 

3d. 

2d. 

1st. 

4. 

two 

four 

eight 

nine 

nine 

5. 

three 

one 

four 

one 

six 

6. 

four 

three 

five 

six 

naught 

7. 

eight 

four 

two 

six 

three 

8. 


one 

naught 

nine 

three 

9. 

seven 

five 

one 

two 

five 

10. 

nine 

four 

nine 

five 

nine 

11. 


one 

four 

nine 

two 

12. 

three 

one 

four 

one 

six 

13. 

two 

nine 

one 

three 

naught 

14. 

one 

six 

naught 

nine 

three 

15. 

nine 

four 

six 

two 

five 

16. 

three 

two 

eight 

naught eight 

17. 

three 

nine 

three 

seven 

naught 

18. 

two 

two 

naught 

four 

six 

19. 

nine 

naught 

eight 

one 

six 

20. 

nine 

one 

four 

three 

nine 

All the figures except naught are called significant figures 

or digits. 

’Naught, called also zero or cipher, 

means none 

and is used when i 

no significant figure 

for an order is given 

7. Write the following numbers in figures: 


Orders. 

4th. 

3d. 

2d. 

1st. 

i. 



one 

five 

none 

2. 



four 

none 

five 

3. 


one 

six 


seven 

4. 


five 

two 

eight 


5. 


four 



four 


6. five in 5th order, eight.in 4th. 

7. two in 3d order, five in 1st. 

8. four in 4th order, four in 2d. 

9. seven in 5th order, three in 3d. 
10. nine in 3d order, one in 1st. 


THE ARABIC SYSTEM. 


5 


11. two in 8d order. 

12. five in 2d order. 

13. three in 5th order. 


14. six in 4th order. 

15. nine in 8d order. 

16. nine in 4th order, 


17. two in 5th order, nine in 4th, two in 1st. 

18. five in 5th order, nine in 4th, three in 3d 

19. five in 6th order, four in 4th, two in 2d. 

20. six in 6th order, five in 5th, two in 2d. 

21. four in 7th order, two in 4th, one in 1st. 


8. Numeration. 

Each order is given a name for the same reason that all 
objects are named—so that we may talk and write about 
them. The 1st order is called units, the 2d tens, the 3d 
hundreds, the 4th thousands. 

Orders. 4th 3d 2d 1st. 

Number. 14 9 2 

Names. Thousands Hundreds Tens Units 

9. A number is read by naming its orders, beginning on the 
left. 

1492 is 1 thousand 4 hundreds^ tens 2 units. 

The name units is always omitted, and usage requires the 
following irregularities: 

Instead of reading 11, one ten one; 12, one ten two; 13, 
one ten 3, etc., we say 


Numbers. 11 
Names. Eleven 

Numbers. 16 

Names. Sixteen 


12 13 

Twelve Thirt 


14 15 


Seventeen 


17 


Thirteen Fourteen 

18 

u Eighteen N 


Nineteen 


Fifteen 

19 



6 


THE 20th CENTURY ARITHMETIC. 


Also, instead of reading 20, two tens; 80, three tens; 40, 


four tens, 

etc., we say 



Numbers. 

20 30 

40 

50 

Names. 

Twenty Thirty 

Forty 

Fifty 

Numbers 

70 

80 

90 

Names. 

Seventy 

Eighty 

Ninety 

10. 

Exercises, 



1 . Write all the numbers to twenty, and name them over 
till you know them perfectly. 

2. 21 is read twenty-one; 22 is read twenty-two; 23, twenty- 
three. Read 24, 25, 26, 27, 28, 29. 

3. Write the numbers from 30 to forty and name them. 

4. Count from forty to sixty. 

5. Name all the tens. 


Express in words: 

6. 92 37 27 

7 . 65 99 43 

8. 76 51 14 

9. 80 29 25 

10. 16 42 39 


Write with figures: 


11. Forty-six. 

12. Sixty-one. 

13. Seventy-six. 

14. Seventeen. 

15. Ninety-two. 


16. Eighty-four. 

17. Thirty-eight. 

18. Twenty-five. 

19. Sixty-six. 

20. Fifty-five. 


11. Hundreds. 

It is the general custom to say and between hundreds and 
tens of each period, but the better way is to omit it. In 
reading decimals (Art. 161), and is used for the separatrix. 
137 is read, one hundred thirty-seven. 


Read: 



Write: 

a. 

b. 

c. 


l. 492. 

225. 

781. 

Two hundred forty-six. 

2. 365. 

415. 

438. 

Four hundred sixty-one. 





THE ARABIC SYSTEM. 


7 



a. 

b. 

c. 

3. 

776. 

865. 

962. 

4. 

280. 

607. 

784. 

5. 

216. 

788. 

685. 

6. 

587. 

898. 

589. 

7. 

199. 

533. 

942. 

8. 

451. 

542. 

864. 

9. 

929. 

325. 

769. 

10. 

742. 

243. 

847. 


Nine hundred seventy. 
Three hundred thirty-nine 
Two hundred fifty. 

Five hundred twenty-six. 
Eight hundred ninety-one. 
Seven hundred seventy-six. 
Six hundred six. 

One hundred one. 


12 . 


Thousands. 


1492 is read, one thousand four hundred ninety-two. 



Read: 

a. 

b. 

l. 

1776. 

12632. 

2. 

1865. 

25491. 

3. 

1733. 

16326. 

4. 

5280. 

34795. 

5. 

2533. 

75534. 

6. 

3542. 

61863. 

7. 

4325. 

45947. 

8. 

6243. 

84178. 

9. 

9181. 

59459. 

10. 

7555. 

99682. 


Write: 


Four thousand three hundred seventy-one, 
Eight thousand seven hundred fifty-four. 
One thousand one hundred eleven. 

Five thousand four hundred eighty-two. 
Two thousand eight hundred thirty-five. 
Nine thousand two hundred sixty-seven. 
Six thousand five hundred ninety-three. 
Three thousand nine hundred nineteen. 
Twelve thousand three hundred forty-six. 
Seven thousand six hundred eighteen. 


13. The following table shows all the orders in practical 
use. This is the hey to numeration. Memorize the names of 
the orders and periods. H. stands for Hundred. 




8 


THE mh CENT URY A RITHME T1C. 

Numeration Table. 


00 1 

« 

O 

f Number. 

5t,h. 

4th. 

3d. 

2d. 1st. 

« j 
£ 

Name. 

Trillions. 

Billions. 

Millions. 

Thousands. Units 



Name. 

Number. 


GO 3 

a o 
o id 


d d 

CD * 


GO d 

d o 
o id 


pq 


1_I 

W H 


go nb 
"d d 


c3 GO 
cj 

» g d 

d 2 d 
Ofl to 


^ d o 

i-J ® 

Mhh 


CD 

d GO -+-» 

5 g'3 

^ H P 


16th 14th 13th, 12th llth 10th, 9th 8th7th, 6th 5th 4th, 3d 2d 1st. 


Each group of three orders, beginning with the 1st order, 
is called a period. The periods of a number are separated 
from each other by commas. The period on the left may 
contain only .one or two orders. The names of the periods 
above trillions in their order are quadrillions, quintillions, 
sextillions, septillions, octillions, nonillions, decillions, etc. 

14. A number is read by naming its orders , beginning on the 
left. Usage, however, requires us to say but once for each 
period that part of the name which is the same. 


125,000 is one hundred thousand twenty thousand five thousand, 
but usage shortens it to one hundred twenty five thousand. 


Read: 

a. 

b. 

c. 

1. 

241,000. 

92,000,000. 

92,241'000. 

2. 

825,000. 

76,000,000. 

76,325,000. 

3. 

18,000. 

42,000,000. 

42,018,000. 

4. 

11,000. 

143,000,000. 

143,011,000. 

5. 

183,000. 

365,000,000. 

365,183,000. 










THE ARABIC SYSTEM. 


9 


15. Rule for numeration: Separate the figures into 
periods. Beginning on the left , read each period. 

62,622,250 is read sixty-two million six hundred twenty- 
two thousand two hundred fifty. 



Read: 


Write: 

Read: 

1 . 

5,280 

6. 

241 thousand n. 

265,780 

2. 

81,416 

7. 

825 thousand 12 . 

3,141,592 

3. 

48,560 

8. 

18 thousand 13. 

3,566,480 

4. 

24,899 

9. 

11 thousand 14. 

62,622,250 

5. 

84,268 

10. 

183 thousand 15 . 

1,479,488,675 

16. 

When 

1 is in 

the 4th order it is usually read with 


the hundreds. Other figures and orders may be similarly 
read. 


1492 is read fourteen hundred ninety-two. 
5280 is read fifty-two hundred eighty. 



Read: 

Write: 

1 . 

1776 

11. Sixteen hundred sixty-four. 

2. 

1066 

12. Seventeen hundred twenty-eight. 

3. 

1865 

13. Fifteen hundred sixty-five. 

4. 

1607 

14. Nineteen hundred. 

5. 

1733 

15. Eighteen hundred forty-five. 

6. 

1893 

16. Seventeen hundred sixty. 

7. 

2533 

17. Twenty-seven hundred forty. 

8. 

3542 

18. Fifty-two hundred. 

9. 

4325 

19. Seventy-five hundred thirty-one. 

10. 

6243 

20. Ninety-three hundred eighty-two. 


17. REVIEW. 

1. What is a number? What is arithmetic? 

2 . What are the systems of notation and their uses? 



10 THE 20th CENTURY ARITHMETIC. 

3. Of what use are numbers? What are consecutive 
numbers ? 

4. Give another name for the Arabic system. What is 
its scale ? 

5 . Name the orders from units to billions. 

6. Name the periods from trillions to units. 

7. Give the rule for writing a number. How are the or¬ 
ders numbered ? 

8. Give the rule for numeration. 

Read the following sentences: 

9. Columbus discovered America in 1492. 

10 . The first English settlement was made at Jamestown 
in 1607. 

11 . Georgia was settled at Savannah in 1733. 

12. There are 5280 feet in a mile. 

13. The Ferris Wheel at The World’s Fair was 265 feet high 
and could carry 1368 people. 

14. The Washington Monument is 555 feet high and the 
Eiffel Tower at Paris is 975 feet high; the great Watkin Tower 
in England will be 1150 feet high. 

15. It is 24,899 miles around the earth. 

16. In an acre there are 43,560 square feet. 

17. The moon is about 238,800 miles from us. 

18. Light travels 186,500 miles in a second and it takes 
493 seconds for the sun’s light to reach the earth. 

19. The sun is about 92,380,000 miles from the earth. 

20. The Bible contains 66 books, 1,189 chapters, 31,173 
verses, 773,746 words, and 3,566,480 letters. The word and 
occurs 46,277 times. 

21. The population of the U. S. in 1890 was 62,622,250, 
consisting of 32,067,880 males and 30,554,370 females. 


THE ARABIC SYSTEM. 


11 


Write the numbers in the following sentences: 

22. St. Augustine was founded by the Spaniards in fifteen 
hundred sixty-five. 

23. The area of Georgia is fifty-eight thousand square 
miles. 

24. The population of Chicago in 1890 was one million 
ninety-nine thousand eight hundred fifty. 

25. By the census of 1890, the population of the United 
States was sixty-two million six hundred twenty-two thou¬ 
sand two hundred fifty. 

26. The population of the world in 1890 was estimated to 
be one billion four hundred seventy-nine million four hun¬ 
dred eighty-eight thousand six hundred seventy-five. 

27. In 1894 the Southern States raised nine million thirty 
five thousand three hundred seventy-nine bales of cotton. 
The largest crop ever produced. 

18. Read: 


l. 25,004 

6. 

62,622,250 

2. 42,070 

7. 

92,880,000 

3. 806,400 

8. 

1,479,488,675 

4. 984,002 

9. 

8,701,025,111 

5. 101,210 

10. 

47,842,600,018 

Write in figures: 




1. Sixteen thousand five. 

2. Eight hundred thousand seventy-three. 

3. Three million eleven thousand seventeen. 

4. Twenty million fifteen thousand one. 

5. One million one hundred one thousand four hun¬ 
dred three. 

6. 18 million 147 thousand 218. 

7. 62 million 622 thousand 250. 

8. 142 million 7 thousand 15. 


12 


THE 20th CENTURY ARITHMETIC. 


9. 5 billion 17 million 107 thousand 14B. 

10. 27 billion 437 thousand 101. 

11. 5 thousand 280. 

12. 314 thousand 159. 

13. 2 million 2 thousand 2. 

14. 50 million 50 thousand 50. 

15. 250 million 250 thousand 250. 

16. Which is the larger number, 41 or 14? 

17. Arrange the figures 1, 4, 9, 2 to make the largest 
number possible. Is it not 9421? Arrange them to make 
the smallest number possible. Is it not 1249? 


Make the largest and smallest 
figures: 

18. 1, 7, 7, 6. 

19. 5, 2, 8, 1. 

20. 3, 1, 4, 5. 

21. 6, 7, 8, 9. 


possible numbers, using all the 


22. 1, 6, 0, 7. 

23. 5, 3, 5, 6, 7. 

24. 4, 9, 7, 2, 1. 

25. 2, 0, 5, 3, 6. 


THE ROMAN SYSTEM. 

Note.—I n inserting the explanation of this system here, the author 
assumes that the pupils know a little addition and subtraction. 
Should they not, its study must be postponed, till after addition and 
subtraction have been studied. 

20. This system employs the following letters: 

Letters. I V X L C D M 

One Five One 

Values. One Five Ten Fifty hundred hundred thousand 

This is the key to the system: commit it to memory. 

These letters may be combined to represent any number 
There are four principles of combination. 



THE ROMAN SYSTEM. 


IB 


I .—Repeating a letter repeats its value. 

A letter is not usually repeated more than three times, though 4 
on time-pieces is written IIII. V, L, and D are never repeated ; VV 
would mean X, LL would mean 0, and DD would mean M. 

= means equals. 

XX=two tens=20. CCC=three hundred=800. 



Read: 


Write: 



a. 

b. 

c. 

d. 

1 . 

XX 

CCC 

two 

three thousand 

2. 

II 

XXX 

thirty 

three hundred 

3. 

cc 

MMM 

three 

two thousand 

4. 

MM 

III 

twenty 

two hundred 


II. — When a letter is placed after one of greater value , their 
values are added. 


VI— 5 and 1=6. LX=50 and 10=60. DC—600. 



Read: 



Write: 



a. 

b. c. 

d. 

e. 

f. 

5. 

VI 

XV CX 

DL 

51 

105 

6. 

XI 

LV DX 

ML 

6 

510 

7. 

LI 

CV MX 

CL 

15 

60 

8. 

Cl 

DV LX 

MC 

11 

55 

Combining I. and II. we have: 




VII- 

=5 arid 2=7. 

LXXX: 

=50 and 80=80. 


Read: 



Write: 



a. 

b. 

C. 

d. 

e. 

9. 

XII 

LXX 

MDCC 

12 

1700 

10. 

CII 

MXX 

MDXX 

58 

1520 

11. 

LIII 

CXX 

MDII 

102 

1502 

12. 

DXX 

MCC 

DCCC 

70 

800 


14 


THE goth CENTURY ARITHMETIC. 


III. — When a letter is placed before one of greater value, its 
value is subtracted from that of the greater. 


IV 

=1 from 

5=4. 

XL=10 from 

50=40. 


Read: 


Write: 



a. 

b. 

c. 

d. 

13. 

IV 

XL 

90 

400 

14. 

IX 

XC 

49 

9 

15. 

IL 

CD 

4 

40 


A letter placed between two of greater value is taken with 
the right-hand one according to III. 

XIV=X and IV ( not XI and V)=14. 



Read: 



Write: 


a. 

b. 


c. d. 


16. XIX 

DXL 


14 59 


17. LIX 

CIX 


19 540 


18. LIV 

MOD 


54 104 

IV.- 

—A bar over a letter, or 

combination of letters, increases 

its value a thousand times. 




V=5 thousand. 

IX= 

9 thousand. 

Read: 


Write: 



a. b. 

c. 

d. 

e. 

19. 

X IV 

C 2 thousand 

10 thousand 

20. 

LT XL 

XV 7 thousand 

17 thousand 

21. 

The following table is 

arranged decimally: 


Thousands. Hundreds. 

Tens. 

Units. 

l. 

M 

C 

X 

I =1111 

2. 

MM 

CC 

XX 

II =2222 

3. 

MMM 

ccc 

XXX 

III =3388 

4. 

IT 

*CD 

XL 

IV =4444 


*400 is also written COCO. 






THE HOMAN SYSTEM. 


15 

5. 

V 

D 

L 

V 

=5555 

6. 

VI 

DC 

LX 

VI 

=6666 

7. 

VTT 

DCC 

LXX 

VII 

=7777 

8. 

VIII 

DCCC 

LXXX 

VIII 

=8888 

9. 

IX 

CM 

XC 

IX 

=9999 


Any number less than ten thousand may be written by using this 
table. There is no Roman character to represent none. 

Roman. M CD XC IX =MCDXCIX 

Arabic. 1 thousand 4 hundred 9 tens 2 units=1492 


Write in figures: Write in letters: 



a. 

b. 

c. 

d. 

e. 

10. 

MC 

DLXII 

MCDXCII 

1565 

1900 

n. 

DVI 

MDC 

MDCCLXXVI 

1738 

1845 

12. 

IV 

CXVI 

MDCCCLXV 

1066 

1607 

13. 

CCL 

MCCL 

CMXCIX 

1815 

1492 

14. 

XCV 

MDCC 

MDCLXVI 

1890 

1776 


21a. REVIEW. 


1. Give the four principles of combination used in the Roman 
notation. 

a table of Roman notation decimally to ten 


2. Write 
thousand. 

Read the following: 

3. Pope Leo XIII. 
King Louis XIV. 
King Henry VI. 
Pope Pius IX. 


7. 

8 . 
9. 

10 . 


Exodus XX. 12. 
Proverbs VIII. 11. 
Proverbs XXIII. 12. 
Psalms XXXII. 8. 


4. 

5. 

6 . 

11 . XLVII. men in 8 years translated the King James I. 
edition of the Bible. The title page of the first one could 
have borne the date MDCX. 




16 


THE 20th CENTURY ARITHMETIC. 


12. The first book printed in England with a date was 
printed by William Caxton in the year MCDLXXVII. 

13 . The inscription on the Liberty Bell was taken from 
Leviticus xxv. 10. 

14 . The date on the Liberty Bell is MDCCLIII. 


Diversions. 

15 . Take one from nine and leave ten. IX; X. 

16. Add one to nine and make twenty. IX; XX. 

17. Take one from nineteen and leave twenty. XIX; XX. 

18. Add one to twenty and make nineteen. XX; XIX. 

19. Take nine from six, ten from nine, and fifty from forty 
and six will remain. 

SIX IX XL 

IX X L 

S I X 

20. To fifty add zero, then five, then a fifth of eight, -and 
the total is the sum of human happiness. LOVE. 


22. REDUCTION. 

IIIII IIIII III • Here are thirteen units. Write the 
number, 13; it represents one ten, three units. 13 units is 
equivalent to 1 ten, 3 units. This process of .changing the 
name but not the value is called reduction. 

///// ///// ////• How many units? Write the number, 
14; read it as units; as tens and units. Read 25 units as 
tens and units. How many tens in the number 45? How 
many tens in 130? Are there not 13 tens? How many units 
in 15 tens? How many tens in 150 units? 




REDUCTION. 


17 



Write the number which expresses how many units in 
each. How many tens are in each? Notice that the left- 
hand figure in the number of units expresses the number of 
tens. Also, that 10 units make 1 ten. 



157 units. 

15 tens 7 units. 

1 hundred 5 tens 7 units. 


18 


THE 20th CENTURY ARITHMETIC. 


22a. The first four orders are Th. H. T. U. Where there 
are more than 9 of any order, the left-hand figure is the next 
higher order. 


In 92 units there are 9 tens. 

In 49 tens there are 4 hundreds. 

In 14 hundreds there is 1 thousand. 

1 . How many tens in 92? 45? 96? 82? 58? 

2 . How many hundreds in 49 tens? 21 tens? 48 tens? 
75 tens? 18 tens? 25 tens? 67 tens? 

3. How many thousands in 14 hundreds? 69 hundreds? 
82 hundreds? 48 hundreds? 91 hundreds? 

4 . How many units in 845? how many tens? 

5 . How many tens in 1492? how many hundreds? 

Fill out the blanks: 


Units. 


T. 

U. 


Tens. 


6. 13 

is 

1 

3 

n. 

75. 

is 

7. 25 

is 



12. 

43 

is 

8. 65 

is 



13. 

80 

is 

9. 92 

is 



14. 

91 

is 

10. 37 

is 



15. 

56 

is 


H. 

7 


T. 

5 


U. 

0 


16. 

17. 

18. 

19. 

20 . 
21 . 
22 . 


T. U. 

42 is 8 tens and how many units? 3 12 

23 is 1 ten and how many units? 

56 is 4 tens and how many units? 

42 is 3 tens and how many units? 

65 is 5 tens and how many units? 

Write the number which has 3 tens, 2 units. Ans. 82. 
Write the number which has 1 ten, 15 units. Ans. 25. 


Write the number which has 



T. 

U. 


T. 

U. 


T. 

U. 

23. 

3 

12 

26. 

5 

13 

29. 

5 

10 

24. 

4 

13 

27. 

8 

12 

30. 

9 

10 

25. 

1 

19 

28. 

1 

15 

31. 

9 

15 


THE DECIMAL SYSTEM. 19 



H. 

T. 

U 


H. 

T. 

U. 


H. 

T. 

u. 

32. 

3 

1 

15 

35. 

3 

12 

5 

38. 

4 

11 

13 

33. 

5 

3 

12 

36. 

4 

17 

0 

39. 

6 

13 

19 

34. 

4 

8 

12 

37. 

5 

10 

0 

40. 

9 

9 

10 


Th. 

H. 

T. 

U. 




Th 

. H. 

T. 

u. 

41. 

1 

4 

8 

12 



51. 


. 1 

32 

0 

42. 

1 

6 

16 

16 



52. 

2 

37 

0 

0 

43. 

5 

2 

7 

10 



53. 

3 

22 

7 

10 

44. 

1 

5 

9 

17 



54. 


51 

18 

0 

45. 

3 

21 

6 

24 



55. 


16 

16 

16 

46. 


14 

25 

13 



56. 


9 

9 

10 

47. 

3 

21 

6 

24 



57. 

5 

9 

10 

0 

48. 

5 

10 

2 

10 



58. 


12 

29 

2 

49. 

1 

3 

18 

12 



59. 

3 

25 

10 

17 

50. 



6 

25 



60. 

4 

31 

42 

10 

Combine 

(add) the following, 

writing as 

one 

number: 


400, 2, 1000, and 90. Ans. 1492. 






61. 

20 and 5. 




66. 

300, 60, 

and 5 



62. 

7 and 40. 




67. 

200, 

5000, and 80. 


63. 

80 and 1. 




68. 

1700, 70, and 

6. 


64. 

9 and 30. 




69. 

92, 

400, and 

1000. 


65. 

4 and 70. 




70. 

700, 

28, 

and 

1000. 



71. 170000, 1, and 8900. 

72. 100000, 500, and 86000. 

73. 8600000, 160, 7, and 68000. 

74. 62000000, 250, 600000, and 22000. 

75. 2300000, 90000000, and 80000. 


23 . Scale of the Arabic Notation. 

A unit is one; as, one apple, one dollar, one. 
Orders: 5th. 4th. 3d. 2d. 1st. 

2 7 5 4 3 



20 


THE mh CENTURY ARITHMETIC. 


In the number 27543 there are three units of the first 
order; 4 units of the second order; how many units of the 
third order? of the 4th? of the 5th? 

The largest number of units of any order that can be ex¬ 
pressed by a single figure is nine. How, then, can we ex¬ 
press more than 9 units of any order? The answer is, we 
cannot express it with one figure, so we always reduce it to 
units of the next higher order. 

Ten units of any order make one unit of the next higher order. 

This principal is used in adding, subtracting, multiplying, 
and dividing numbers. 


24. ADDITION. 

/////. Here are 5 marks, make 3 more and count as you 
make them: ////// 6; ///// // 7; ///// /// 8. 

You have found out by counting that the sum of 5 and 3 
is 8. This is the only way to find the sum of two digits, but 
by remembering the sums of every pair of digits, there is a 
process called addition by which the sum of any two num¬ 
bers may be found. 

The sign of addition, +, is called plus. 

5 + 3 = 8 is read 5 plus 3 equals 8. 

1 . If you make 5 marks and then 4 more, how many will 
there be? 

2. John put in his bank 7 cents at one time and 8 cents 
at another; how many cents did he put in? 

3 . Roy rode 9 miles on his bicycle in the morning and 8 
miles in the afternoon; how far did he ride during the day? 



ADDITION. 


21 


4 . How many are 8 apples and G apples? 

5 . How many are 5x and 7x ? Make 5 x’s, then 7 more, 
and count how many in all. 

Adding is performed oftener than any other operation in arithme¬ 
tic; it should, therefore, be very thoroughly learned. The name of 
a familiar object, a book, for instance, comes to the mind at once, 
without effort, on seeing it; so ought the sum of every pair of digits 
on seeing them. Much practice, becoming familiar with them, is the 
only way of obtaining this result. 

Memorizing the sums of every pair of digits is a very great task. 
Memory is strengthened by attention and by repetition. Attention can 
be obtained by making the subject attractive and this will depend on 
the genius of the teacher. Pupils should not be required to mem¬ 
orize many sums each day and they should be required to find the 
sums by counting. They will then appreciate the value of the num¬ 
ber and also the labor saved by memorizing the sums. Let pupils 
make diagrams similar to those below and fill in the blank spaces. 
Attention should be constantly called to the commutative law of ad¬ 
dition ; example, if 9 —(— 5 == 14, then also 5 —|— 9 — — 14. 

Perfection in addition must be attained sooner or later, but this 
perfection must not be striven for by excluding all other ideas 
and operations. The repetition of addition in other operations will 
eventually fix it indelibly on the mind of the pupil. 



7 

2 5 

3 

3 




5 


7 


2 



5 

6 


| 




7 

4 

8 

6 

9 

7 


■11 




8 





17 

4 



12 



9 







Fill out the blanks. 


































22 


THE goth CENTURY ARITHMETIC. 





The Arabic method of notation is generally believed now to have 
been invented by the Hindoos. 

The nine digits whose origin has been the subject of much research 
have been attributed by some able scholars to the initial letters of the 
Sanscrit numerals in analogy to the Grecian and Roman method of 
representing numbers by initial letters. This is the most recent and 
probable theory. 


























ADDITION. 


28 



The great Italian poet, Petrarch, has left us the oldest authentic 
date written in numeral characters: the date is 1375, written upon a 
copy of St. Augustine. The earliest inscription on a monument in 
these characters in England is in the Church of Ware on a brass plate 
commemorating the death of Ellen Wood 1454 

Euclid was the first writer on arithmetic whose writings have come 
down to us. 

Pythagoras lived about 600 B. C. He is the reputed inventor of 
the multiplication table. 





























24 


THE SOth CENTURY ARITHMETIC. 


The following combinations are the foundation of all 


arithmetical operations. 

The sum less than 10: 

1 2 8 2 4 

] 1 1 _ 2 _ 

5 4 7 6 5 

2 _ 8 _ 1 _ 2 _ 8 _ 

The sum 10 or more: 

9 8 7 6 5 9 

1 2 8 4 5 2 


8 5 4 3 6 

2 i 2_ 8^ _1 

4 8 7 6 5 

4 12 3 4 


8 7 6 9 8 7 

8 4 5 8 4 5 


698798798989 

645656767788 


Addition Table. 



1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

2 

3 

4 

i 5 

6 

7 

8 

9 

10 

11 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 


This table is for reference from the exercises below. The teacher 
should explain how to use the table. 


































ADDITION. 


25 


Point between the following digits and announce the sum 
of the two digits between which the pointer is. 

6 , 2 , 2 , 3 , 9 , 5 , 6 , 6 , 3 , 8 , 8 , 2 , 5 , 7 , 6 , 9 , 9 , 2 , 7 . 
5 , 8 , 9 , 7 , 7 , 3 , 5 , 5 , 4 , 8 , 7 , 4 , 4 , 6 , 8 , 2 , 4 , 3 , 3 . 

26 . 

1. Add 4 to 8, to 18, to 28, to 88, to 43, to 53. 

2. Add 5 to 7, to 17, to 27, to 37, to 47, to 57. 

3. Add 6 to 5, to 15, to 25, to 35, to 45, to 55. 

4. Add 7 to 2, to 12, to 22, to 32, to 42, to 52. 

5. Add 8 to 4, to 14, to 24, to 34, to 44, to 54. 

6. Add 9 to 6, to 16, to 26, to 36, to 46, to 56. 


Add by 2’s from 2 to 16, 

thus: 2, 4, 

6, 8 

;, 10, 12, 

T. 

Add 

by 

2’s 

from 

1 

to 

21, 

thus: 1 

,1 

M 

5, 7, etc. 

8. 

Add 

b y 

8’s 

from 

1 

to 

31. 

From 

2 

to 

32. 

9. 

Add 

by 

3’s 

from 

3 

to 

33. 

From 

4 

to 

34. 

10. 

Add 

by 

4’s 

from 

1 

to 

41. 

From 

3 

to 

43. 

11. 

Add 

by 

4’s 

from 

2 

to 

42. 

From 

4 

to 

44. 

12. 

Add 

by 

5’s 

from 

1 

to 

51. 

From 

4 

to 

54. 

13. 

Add 

by 

5’s 

from 

3 

to 

63. 

From 

5 

to 

55. 

14. 

Add 

by 

6’s 

from 

1 

to 

61. 

From 

4 

to 

64. 

15. 

Add 

by 

6’s 

from 

2 

to 

62. 

From 

5 

to 

65. 

16. 

Add 

by 

6’s 

from 

3 

to 

63. 

From 

6 

to 

66. 

17. 

Add 

by 

7’s 

from 

1 

to 

71. 

From 

4 

to 

74. 

18. 

Add 

by 

7’s 

from 

2 

to 

72. 

From 

5 

to 

75. 

19. 

Add 

by 

7’s 

from 

3 

to 

73. 

From 

6 

to 

76. 

20. 

Add 

by 

8’s 

from 

1 

to 

81. 

From 

2, 

to 

82. 

21. 

Add 

by 

8’s 

from 

3 

to 

83. 

From 

4 

to 

84. 

22. 

Add 

by 

8’s 

from 

5 

to 

85. 

From 

7 

to 

87. 

23. 

Add 

by 

9’s 

from 

1 

to 

91. 

From 

6 

to 

96. 

24. 

Add 

by 

9’s 

from 

2 

to 

92. 

From 

7 

to 

97. 

25. 

Add 

by 

9’s 

from 

5 

to 

95. 

From 

9 

to 

99. 


26 


THE 20th CENTURY ARITHMETIC. 


27. I. —Add each of the following columns twice, beginning 
first at the bottom then at the top. Compare the two sums 
found for each column. 

In adding , name results only. Thus, in example 1, say 7, 9, 
14, 17, 19; do not say 7 and 2 are 9, and 9 and 5 are 14, etc. 


1 . 2 . 3 . 4 . 5 . 

2 4 6 8 8 

8 7 8 9 2 

5 1 8 2 7 

2 9 8 7 8 

7 4-2 6 2 


6 . 7 . 8 . 9 . 10 . 11 . 12 . 

5 7 9 4 5 6 7 

1 4 8 5 4 6 2 

6 5 9 8 6 2 5 

4 9 2 6 5 8 4 

4 8 7 6 5 8 4 


13 . '4 — 8 — (— 3 — (— 7 + 2 — 

14 . 5 + 94-4 + 8 + 8 = 



17. 3 + 2 + 7 + 9 + 3 + 8 = 

18 . 5 + 44 5+8 - r 5 + 6 = 

19 . 7 + 8 + 3+ 6+ 7 + 4 == 


2 = 


II.—Add the following columns first up, then down; and 
the rows first from the left, then from the right. 


27 . 

28 . 

29 . 

21 . 4 

4 

4 

22. 7 

2 

3 

23 . 3 

8 

9- 

24 . 5 

5 

5 

25 . 8 

7 

5 

26 . 4 

6 

3 

43 . 2 

5 

6 

44 . 4 

7 

6 

45 . 7 

8 

0 


30 . 31 . 38 . 

4 4 32 . 9 

5 9 33 . 7 

2 6 34 . 3 

5 5 35 . 2 

9 4 36 . 6 

6 7 37 . 4 


9 4 2 3 
18 5 3 
3 12 4 


39 . 40 . 41 . 42 . 

4 3 7 6 

7 7 7 7 

5 7 9 6 

3 4 5 6 

6 6 6 6 

5 2 8 6 


6 


7 
0 

8 


4 

6 


5 

2 

5 



ADDITION. 


27 


56. 57. 58. 

46. 6 7 3 

47. 5 1. 7 

48. 3 8 6 

49. 0 3 7 

50. 4 2 4 

51. 2 7 7 

52. 9 0 2 

53. 8 9 9 

54. 6 4 2 

55. 3 2 5 


59. 60. 61. 62. 63 

1 8 9 7 6 

4 2 7 8 1 

8 5 3 1 9 

9 9 6 4 0 

6 2 0 7 5 

8 5 5 1 8 

5 9 4 7 2 

8 6 0 3 9 

8 3 9 7 2 

1 6 4 7 9 


64. 65 

4 9 

8 2 

2 3 

0 3 

9 4 

6 8 

.9 7 

3 0 

1 4 

2 8 


Frequent practice on these exercises will give the ability to add 
rapidly and accurately. 

DEFINITIONS. 

Addition is the process of finding one number which con¬ 
tains as many units as two or more numbers. 

The numbers added are called addends. 

The result of addition is called the sum or amount. 

Only like numbers can be added. 


28. Adding Numbers Larger Than 9. 

I.—When the sum of each column is less than 10. 

A farmer sold some corn for 150 dollars, wheat, for 223 
dollars and cotton for 415 dollars. How many dollars did 
he receive? 


p d • CO 

fl S.-S 


$788 


The dollar sign, $, is placed to the left of the 
number. It should be used with the first addend 
only. 

Directions. —Write the numbers so that the 
same orders shall be under each other. Begin 
on the right and add each column. 






28 


THE goth CENTURY ARITHMETIC. 


1 . 

2. 

3. 

4. 

5. 

6. 

$217 

141 a 

408 

$411 

206 x 

365 

431 

218 a 

270 

263 

340 x 

402 

250 

300 a 

311 

112 

231 a 

221 


7. A farmers land is worth $2180, his house $1200, and 
his stock $400. What is it all worth ? 

8. Three bales of cotton weigh respectively 424, 458, and 
510 pounds. How many pounds in all ? 

9 . If Santa Claus has 820 dolls, 303 drums, and 314 
books, how many presents has he in all ? 

10. How many parents does every boy have ? In a school 
there are 213 boys and 211 girls, none of whom are brothers 
or sisters. How many parents altogether have these chil¬ 
dren? 

11. Solomon’s temple was built 1014.years before Christ. 
If it were still standing how old would it be 1900 years after 
Christ ? 


12. 

13. 

14. 

15. 

16. 

17. 

1492 

1243 

1212 

1776 

3146 

3116 

5200 

6421 

3010 

5112 

3210 

4532 

3106 

1032 

3146 

4001 

2112 

2121 


II.—When the sum of any column is more than 9. 

Ten units of any order make one unit of the next higher order. 

Th. H. T. U. 

13 units = 13 

25 tens = 2i 5 

14 hundreds = 1 4 

Therefore, from the nature of the decimal system, when 
the sum of any column is more than 9, the left figure of the 
partial sum is units of the next higher order. 











ADDITION. 


29 


Find the sum of 385, 482, and 796. 

Directions. —Write the numbers so that 
21 the same orders will fall under each other. 

' ~~ Draw a line above and below. Begin on 
380 the right and add each column. If the 
482 sum of any column (except the extreme 
796 left one) is more than 9, place the units of 
- - the sum under the column and the tens above 

1663 the next column to the left. Add the 
number above the column with the column 
over which it is placed. The work on the 
right is to explain that on the left. 

To detect mistakes in addition, it is well to add each col¬ 
umn twice: from the bottom up, and from the top down. 

29 . 1 . In an example in addition, the sum of the units 

column is 52, of the tens, 24, of the hundreds, 12. What is 
the total sum ? (See Art. 22.) 



Partial 

sums. 


1663 


Find the total sum in each of the following: 


Sum of 

Sum of 

Sum of 

hundreds. 

tens. 

units. 

2. 36 

34 

14. 

3. 30 

25 

5. 

4. 21 

12 

3. 

5. 15 

27 

11. 



Sum of 

Sum of 

Sum of 

hundreds. 

tens. 

units. 

6. 

17 

23 

18. 

7. 

21 

19 

10. 

8. 

10 

16 

24. 

9. 

16 

21 

31. 


In writing numbers, care should be taken to make the figures clear 
and plain, so that 3 will not be mistaken for 5, or 7 for 9.. See, also, 
that all the figures in a column of like orders are placed directly un¬ 
der each other. Always set down the figures to be carried ; then, if 
necessary, any column may be added without the trouble of adding 
the preceding column. 

The ability to add accurately and rapidly is more than worth all 
the pains necessary to acquire it. 


10 . 

Feet. 

1728 

365 

5280 


11. 

12. 

Dollars. 

Bushels. 

449 

684 

876 

365 

984 

492 


13. 

14. 

Pounds. 

Miles. 

3845 

7284 

1309 

9782 

1849 

8447 








30 


THE goth CENTURY ARITHMETIC. 


15 . Four bales of cotton weigh respectively 485, 510, 463, 
and 513 pounds. Find the sum of their weights. 

16. The land surface of the globe is composed of the 
eastern continent, 35,413,799 square miles; the western con¬ 
tinent, 16,237,535 square miles; Australia and the islands, 
3,709,781 square miles. What is the entire land area ? 

17 . Find the sum of the lengths of the following rivers: 
Missouri, 2908; Mississippi, 2616; Arkansas, 2170; Red, 
2100; Nebraska, 1200; Yellowstone, 1000; and Ohio, 975 
miles. (These are the Mississippi and its longest branches.) 

18. In one mile there are 5280 square feet; how many feet 
in five miles ? 


19. Find the area of the United States from the following data. 

Area ceded by Great Britain, 

1783 

830,000 

Louisiana purchase from France, 

1803 

1,182,752 

Florida purchase from Spain, 

1819 

59,268 

Annexation of Texas, 

1845 

274,356 

Treaty with Mexico, 

1848 

522,568 

Texan claims from Mexico, 

1850 

96,707 

Gadsden purchase from Mexico, 

1853 

45,527 

Alaska purchase from Russia, 

1867 

577,390 

Total Area, 

/ 


20 . 21 . 


Estimated population 

Number dollars paid by 


of the world in 1897. 

U. S. in pensions 1890-5. 

Asia, 

825,924,000 

106,493,890 

Europe, 

357,379,000 

118,548,859 

Africa, 

163,933,000 

141,086,948 

America, 

121,713,000 

158,155,342 

Oceanica, 

7,500,400 

140,772,163 

Australia, 

3,230,000 

140,959,361 

Total, 








ADDITION. 


81 



22. 

23. 

Population of cities in 

Number bales cotton 


1890, 1891, or 1892. 

raised in TJ. S. 1886-95. 

London, 

4,281,481 

6,550,215 

Paris, 

2,447,957 

6,513,624 

New York, 

1,801,789 

7,017,707 

Berlin, 

1,579,244 

6,935,082 

Tokio, 

1,389,684 

7,313,726 

Vienna, 

1,364,548 

8,655,518 

Philadelphia, 

1,142,653 

9,038,707 

Chicago, 

1,099,850 

6,717,142 

St. Petersb’g, 

1,035,439 

7,527,211 

Brooklyn, 

957,163 

9,892,766 


Sum, 


* Population of United States in 1890. 



24. 

25. 

26. 

States and 

Total 

r - 

-Sex. -s 

Territories. 

Population. 

Male. 

Female. 

Alabama, 

1,513,017 

757,456 

755,561 

Arizona, 

59,620 

36,571 

23,049 

Arkansas, 

1,128,179 

585,755 

542,424 

California, 

1,208,130 

700,059 

508,071 

Colorado, 

419,198 

245,247 

166,951 

Connecticut, 

746,258 

369,538 

376,720 

Delaware, 

168,493 

85,573 

82,920 

District of Columbia,230,392 

109,584 

120,808 

Florida, 

391,422 

201,947 

189,475 

Georgia, 

1,837,353 

919,925 

917,428 

Idaho, 

84,385 

51,290 

33,095 

Illinois, 

3,826,351 

1,972,308 

1,854,043 

Indiana, 

2,192,404 

1,118,347 

1,074,057 

Iowa, 

1,911,896 

994,453 

917,443 

*The sum of males and females give the total population. 

Use this fact to make 

additional examples. 










32 


THE 20th CENTURY ARITHMETIC. 



27. 

28. 

29. 

States and 
Territories. 

Total 

Population. 

Male. 

-Sex. - s 

Female. 

Kansas, 

1,427,096 

752,112 

674,984 

Kentucky, 

1,858,635 

942,758 

915,877 

Louisiana, 

1,118,587 

559,350 

559,237 

Maine, 

661,086 

332,590 

328,496 

Maryland, 

1,042,390 

515,691 

526,699 

Massachusetts, 

2,238,943 

1,087,709 

1,151,234 

Michigan, 

2,098,889 

1,091,780 

1,002,109 

Minnesota, 

1,301,826 

695,321 

606,505 

Mississippi, 

1,289,600 

649,687 

639,913 

Missouri, 

2,679,184 

1,385,238 

1,293,946 

Montana, 

132,159 

87,882 

44,277 

Nebraska, 

1,058,910 

572,824 

486,086 

Nevada, 

45,761 

29,214 

16,547 

New Hampshire, 

376,530 

186,566 

189,964 


30. 

31. 

32. 

New Jersey, 

1,444,933 

720,819 

724,114 

New Mexico, 

153,593 

83,055 

70,538 

New York, 

5,997,853 

2,976,893 

3,020,960 

North Carolina, 

1,617,947 

799,149 

818,798 

North Dakota, 

182,719 

101,590 

81,129 

Ohio, 

3,672,316 

1,855,736 

1,816,580 

Oklahoma, 

61,834 

34,733 

27,101 

Oregon, 

313,767 

181,840 

131,927 

Pennsylvania, 

5,258,014 

2,666,331 

2,591,683 

Rhode Island, 

345,506 

168,025 

177,481 

South Carolina, 

1,151,149 

572,337 

578,812 

South Dakota, 

328,808 

180,250 

148,558 

Tennessee, 

1,767,518 

891,585 

875,933 

Texas, 

2,235,523 

1,172,553 

1,062,970 

Utah, 

207,905 

110,463 

97,442 




ADDITION. 38 



33. 

34. 

35. 

States and 

Total 

, _ 

-Sex.-—-% 

Territories. 

Population. 

Male. 

Female. 

Vermont, 

332,422 

169,327 

163,095 

Virginia, 

1,655,980 

824,278 

831,702 

Washington, 

349,390 

217,562 

131,828 

West Virginia, 

762,794 

390,285 

372,509 

Wisconsin, 

1,686,880 

874,951 

811,929 

Wyoming, 

60,605 

39,343 

21,362 


Total, 


30 . REVIEW. 

1. What is addition ? 

2. Name the parts in addition: 5 + 3=8. 

3. What is the sign of addition and its name ? 

4. When the addends are 3, 5 and 8, what is the sum ? 

5. Write 1492 five times and find the sum. 

6. A, B, and C began business together. A put in 
$17,500; B $2,550 more than A; and C as much as A and 
B together. What was the whole amount put in ? 

7. How many times must 1760 be used as an addend to 
give 5280 ? 

8. Find the sum of the five largest numbers that can be 
expressed by the figures 8, 7, 6, 3, and 5. 

9 . Find the sum of the five smallest numbers that can 
be expressed by the figures 1, 2, 3, 7, and 8. 

10. A has $776; B has twice that much and C three 
times that much; how much have they all together ? 

11. Use 142,857 as an addend 4 times and find the sum. 

12. Use 142,857 as an addend 5 times and find the sum. 

13. Use 142,857 as an addend 6 times and find the sum. 

14. In one year there are 365 days; how many days in 
5 years ? 








34 


THE mh CENTURY ARITHMETIC. 


15. In one mile there are 1760 yards; how many yards 
are in 5 miles ? 

Diversions. —Add the rows, columns, and two long diag¬ 
onals in the following figures and see if the sums for each 
row, etc., are not the same. 


16. 17. 18. 


3 

16 

9 

22 

15 

20 

8 

21 

14 

2 

7 

25 

13 

1 

19 

24 

12 

5 

18 

6 

11 

4 

17 

10 

23 


1 

16 

11 

6 

13 

4 

7 

10 

8 

9 

14 

3 

12 

5 

2 

15 


4 

9 

2 

3 

5 

7 

00 

1 

6 


31 . 


MULTIPLICATION. 


Ill III III III III • Here are 5 groups of marks with 3 
marks in each group. Add by 3’s and find how many marks 
in all. You have found by adding that 5 times 3 is 15. This 
is the way to find the product of two digits. By remember¬ 
ing the product of every pair of digits, there is a process 
called multiplication by which the product of any two 
numbers may be found. 

The sign of multiplication is X . 

l. 5 X 3 is read 5 times 3. 


How m,any are 

2. .// // f Two times 2 ? 6. .// // 11 // // // f 

3. // //"//? Three times 2? 7. 11 // // // // // // ? 

4. // H II IP 4X2? 8. // // // // // // // // f 

5. // // //////76X2? 9. // // // // // // // // // ? 
































MUL T1 PLICA TION. 


35 


io. a. 

b. 

c. 

d. 

e. 

f. 

g- 

h. 

2 

3 

4 

5 

6 

7 

8 

9 

2 

3 

4 

5 

6 

1-7 

< 

8 

9 

2 

3 

4 

5 

6 

7 

8 

9 


Add these columns and try to remember the sums. 


11. How much is 2 taken 3 times? What is 3 times 7? 
8 times 4 ? 3 times 5 ? 3 limes 9 ? 3 times 6 ? 

If you do not remember these products, add as in Example 10. 


12. a. b. e. 

2 3 4 

2 3 4 

2 3 4 

2 3 4 


d. e. f. g. h. 

5 6 7 8 9 
5 6 7 8 9 
5 6 7 8 9 
5 6 7 8 9 


Add these columns and try to remember the sums. 

13. How much is 2 taken 4 times ? What is 4 times 3 ? 
4 times 7 ? 4 times 5 ? 4 times 9 ? 4 times 4 ? 

Let the pupil construct tables similar to these and thus learn the 
other multiplication tables. Learning them is a simple act of mem¬ 
ory and they should be repeated, little at a time, till memorized well 
enough to proceed. The repetitions of later work will fix them 
indelibly in the mind. 

14. What is the difference between 

3 groups of 4 marks and 4 groups of 3 marks ? 

nn nn nn and /// /// /// /// t 

3 times 4 and 4 times 3 ? 

This law of commutation should be noticed for the product of every 
pair of digits. The labor of learning the multiplication table is thus 
lessened almost one half. 

The product of other digits by 9 are usually the hardest to re¬ 
member. 

Look at the table below and notice that up to 9 X 10 the sum of 
the digits of the products by 9 is always 9, and that the tens of the 
products is always one less than the number multiplied. 



36 


THE 20th CENTURY ARITHMETIC. 





The oldest text-book on arithmetic which employs Arabic figures 
and decimal notation is undoubtedly that of Aricenna, an Arabian 
physician who lived about 1000 A. D. 

The first arithmetic published was in 1478. In 1544, Michael Stifel 
published his “Arithmetic Integra.” He is the acknowledged in¬ 
ventor of the signs -J-, —, and -j/ . 

In 1557, Robert Recoi’de invented and used the sign = in an algebra. 





























MUL TI PLICA TION. 


87 





In 1631, Wm. Oughtred in his “Clavis Mathematicse” first used 
the sign X . 

Dr. John Pell (1610-1658), professor of mathematics and philoso¬ 
phy at Breda, invented 

Descartes invented exponents, ab (— a b) is said to be due to 
Thomas Harriot, 1631. 

Newton invented the binomial theorem. 































88 


THE mth CENTURY ARITHMETIC. 


32 . Multiplication Table. 



2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24. 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9 

18 

27 

36 

45 

54 

63 

72 

81 

90 

99 

108 

10 

20 

1 80 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11 

22 

88 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

86 

48 

60 

72 

84 

96 

108 

120 

132 

144 


This table is for reference. 

Nothing added to itself is still nothing. 

2 X 0 is 0. l. 0 X 4 is what ? 

0 X 2 is 0. 2 . 5 X 0 is what ? 

7 X 0 is 0. 3 . 7 X 0 is what ? 


4 . 8 X 0 =? 

5 . 0 X 9 =? 

6. 0X6—? 


33 . Problems. 

1 . There are 2 pints in 1 quart; how many pints are 
there in 7 quarts ? 


















































MULTIPLICA TION. 


89 


2. If an apple weighs 5 ounces, how much will 9 apples 
weigh ? 

3. There are 8 feet in 1 yard; how many feet in 5 yards? 

4. A book-case has 7 shelves and there are 9 books on 
each shelf; how many books in the case ? 

5. Roy rode 8 miles an hour for 6 hours; how far did 
he go? 

6. There are 6 rows of desks and 8 desks in each row; 
how many desks in all ? 

7. The States in the United States would make 9 rows 
and have 5 States in each row; how many States are there ? 

8. There are 4 quarts in 1 gallon; how many quarts in 
7 gallons ? 

9. 8 silver dollars side by side extend a foot. On a 
square foot could be placed 8 of these rows; how many sil¬ 
ver dollars could be placed on a square foot ? 

10. The number of presidents to 1900 could be put in 9 
groups of 8 each; how many presidents ? 

11. How many days in 8 weeks ? 

12. There are 4 pecks in 1 bushel; how many pecks in 7 
bushels ? 

13. There are 4x7 canals in the United States; how 
many are there ? 

14. The Nicaragua canal will require 9X8 miles of ex¬ 
cavation. 

15. Point between each pair of the following digits and give the 
products of each. 

6, 2, 2, 8, 9, 5, 6, 6, 8, 8, 8, 2, j5, 7, 6, 9, 9, 2, 7, 
5, 8, 9, 7, 7, 8, 5, 5, 4, 8, 7, 4, 4, 6, 8, 2, 4,. 8, 8. 

Learn the multiplication table thoroughly. Practice this exercise 
often. 


40 


THE tOth CENTURY ARITHMETIC. 


DEFINITIONS. 

Multiplication is a short process for finding the sum 
when the same number is repeated. 

The multiplicand is the number repeated. 

The multiplier is the number which shows how many 
times the multiplicand is repeated. 

The result is called the product. 

A concrete number is a number applied to some thing. 
3 marks, 4 apples, 1 horses, 5 x. 

An abstract number is simply a number of units. 6, 9, 5. 


34 . 7* + 7* + 7z = 21a;, 7sX 8 = 21a:. 7 X 3 = 21. 

Only like numbers can be added: the sum of concrete num¬ 
bers is concrete; the sum of abstract numbers is abstract. 
Therefore,, the multiplicand and product must be either both 
concrete or both abstract. The multiplier, however, is 
always abstract, because it only shows how many times the 
multiplicand is used as an addend. 

35 . When the multiplier comes first, the multiplication 
sign is read times; when last, the sign is read multiplied by. 

3 X lx is read 3 times lx. lx X 3 is read lx multiplied by 3. 

When both numbers are abstract, either may be consid¬ 
ered the multiplier. 

Name the parts of 3 X 7x = 21x. Ans. 3 is the multi¬ 
plier, 7x the multiplicand, and 21a; the product. 

36 . Multiply 231 by 3. 


H. T. U. 
2 3 1 
3 

6 9 3" 


Directions. —Place the multiplier H. T. TJ. 

under the units of the multiplicand. 2 3 1 

Multiply each digit of the multipli- 9 Q i 

cand by the multiplier. 3 times 1 A o 1 

unit is 3 units; 3 times 3 tens is 9 tens; 2 3 1 

3 times 2 h-undreds is 6 hundreds. 7> TTYT 





MUL TI PLICA TION. 


41 


1 . 2 . 


3 . 

4 . 

5 . 

341 212 


323 

423 

211 

2 4 


3 

2 

4 

Multiply 





6 . 3122 by 3. 

11 . 

3123 by 3. 

16 . 

3013 by 3. 

7 . 4324 by 2. 

12 . 

2211 by 4. 

17 . 

2423 by 2. 

8 . 1312 by 3. 

13 . 

1111 by 8. 

18 . 

1022 by 4. 

9. 2121 by 4. 

14 . 

3124 by 2. 

19 . 

3141 by 2. 

10 . 3223 by 3. 

15 . 

2112 by 4. 

20 . 

1021 by 5. 


Th. H. T. U. 

5 2 0 7 
3 

1 5 6 2 1 


Explanation. —3 times 7 units is 21 units, 
which is 2 tens 1 unit, (Art. 22.) 3 times 5 

thousands is 15 thousands, which is 5 thou¬ 
sand and 1 ten thousand. 


21. 5207 by 4. 

26 . 

7080 by 9. 

31 . 

4108 by 7. 

22. 6109 by 7. 

27 . 

5070 by 6. 

32 . 

8091 by 9. 

23 . 3208 by 4. 

28 . 

6009 by 8. 

33 . 

7108 by 8. 

24 . 7091 by 8. 

29 . 

4007 by 7. 

34 . 

9009 by 9. 

25 . 3081 by 9. 

30 . 

6071 by 9. 

35 . 

8108 by 8. 

Multiply 1492 by 9. 




1st step. 

2d step. 

3d step. 


4th step. 

1492 

1492 

1492 


1492 

9 

9 

9 


9 

18 

_ 18 

3618 


3618 


81 

81 


981 




Ans. 

13428 











42 


THE 20tli CENTURY ARITHMETIC. 


Explanation. — 1st step: 9 times 2 units is 1 ten, 8 units. 2d step: 
9 times 9 tens is 8 hundreds 1 ten; place the tens under each other. 
3d step: 9 times 4 hundreds is 3 thousands, 6 hundreds; place as shown. 
4th step: 9 times 1 thousand is 9 thousands; place as shown. Add. 
Place like orders in the same column. 


Multiply 

36 . 1492 by 8. 

37. 1492 by 7. 

38 . 1776 by 9. 

39. 1776 by 8. 

40 . 1776 by 7. 

51 . 142,857 

52 . 828,288 
53. 425,846 


41 . 

6485 

by 

6. 

42 . 

7349 

by 

7. 

43 . 

6884 

by 

8. 

44 . 

9447 

by 

9 

45 . 

6553 

by 

5. 

by 8. 



54 . 

by 9. 



55 . 

by 8. 



56 . 


46 . 

7942 

by 

6 

47 . 

6893 

by 

4 

48 . 

7265 

by 

7, 

49 . 

8496 

by 

7 

50 . 

9876 

by 

9 


314,169 by 9. 
391,763 by 8. 
142,857 by 7. 


37. Multiply 1728 by 3. Read Art. 22. 

1728 Directions. —Place the multiplier under units 1728 
q of the multiplicand. Multiply each digit of the 1728 

_ multiplicand by the multiplier. 3 time 8 units 

5184 is 24 units, which is 2 tens 4 units. Write ____ 

down the 4 units and add the 2 tens to the pro- 5184 

duct of tens, etc. 

1 . 2 . 3 . 4 . 

5280 6141 4086 62107 

3 7 9 8 


Multiply 


5 . 

1093 

by 

4. 

13 . 

31,416 

by 

8. 

21 . 

31,416 

by 

7. 

6 . 

365 

by 

7. 

14 . 

63,360 

by 

6. 

22 . 

63,360 

by 

8. 

7 . 

640 

by 

9. 

15 . 

39,370 

by 

9. 

23 . 

39,370 

by 

6. 

8 . 

5280 

by 

6. 

16 . 

21,504 

by 

7. 

24 . 

21,504 

by 

9. 

9 . 

6141 

by 

8. 

17 . 

43,560 

by 

5. 

25 . 

43,560 

by 

7. 

10 . 

4086 

by 

5. 

18 . 

48,667 

by 

4. 

26 . 

48,667 

by 

9. 

11 . 

7925 

by 

7. 

19 . 

46,656 

by 

7. 

27 . 

46,656 

by 

8. 

12 . 

2902 

by 

7. 

20 . 

36,524 

by 

4. 

28 . 

36,524 

by 

7. 








MUL TIPLICA TION. 


43 


29. In every yard there are 8 feet; how many feet are in 
1760 yards ? 

30. Multiply 142,857 by 2, by 3, and by 4. Notice the cu¬ 
rious fact that the products consist of the same figures in the same 
order, beginning at a different place. 

31. Multiply 142,857 by 8. In the product, remove the left 
hand digit and add it to the right hand one ; notice the curious result. 

32. When cotton is worth 8 cents a pound, how much is 
a bale weighing 475 pounds worth ? 

33. Multiply 365 by 2, by 3, and by 5. Add the three 
products and compare it with the multiplicand. Add the 
multipliers 2, 3, and 5. 

34. Sound goes 1093 feet in a second; how far will it go 
in 5 seconds ? The product is 185 feet more than a mile; 
how many feet in a mile ? 

35. The distance around a circle is about 3 times the dis¬ 
tance across it. If it were exactly 3, how far would a 
bicyclist travel in riding 5 times around a circular track 
which is 352 feet across ? 

36. A bicyclist and a pedestrian start at the same time 
from Atlanta towards Chattanooga. The former travels 7 
miles an hour, the latter 3; how far apart will they be in 1 
hour ? When they have traveled 16 hours ? Draw diagram. 

37. Washington was made president in 1789. In what 
year will the 27th presidential term end ? A term is 4 years. 

38. Iron weighs 8 times as much as water. If a certain 
amount of water weighs 41,543 pounds, how much would the 
same amount of iron weigh ? 


38. To multiply by 10, 100, 1000, etc 


Read Art. 22. 


Th. H. T. U. 
5 0 
6 0 
3 0 


10 times 5 Units = 50 units 
10 times 6 Tens = 60 tens = 

10 times 8 Hundreds = 30 hundreds = 3 


44 


THE mh CENTURY ARITHMETIC. 


We see from this that each order is multiplied by 10 by 
moving it one place to the left. 


Th. H. T. U. Th. H. T. U. 

865X 10 = 3 6 50 
365 X 10 X 10 = 365 X 100. 

365 X 10 X 10 X 10 = 365 X 1000. 


To multiply by 10, 100 , 1000, etc., annex as many naughts to 
the multiplicand as are in the multiplier. 


1. 365 X 10. 

2. 1728 X 10. 

3. 144 X 10. 

4 . 62 X 10. 

5. 1492 X 10. 


Examples. 

6. 365 X 100. 

7. 1728 X 100. 

8. 144 X 100. 

9. 62 X 100. 

10 . 1492 X.100. 


11. 365 X 1000. 

12. 1728 X 1000. 

13 . 144 X 1000. 

14 . 62 X 1000. 

15 . 1492 X 1000. 


16 . A keg of nails weighs 100 pounds. What is the weight 
of 1728 kegs of nails ? 

17 . There are 100 years in a century. How many times 
has the earth gone round the sun in 19 centuries ? 

18 . A bicyclist and a pedestrian travel in opposite direc¬ 
tions from Atlanta at the rate of 7 and 3 miles an hour 
respectively. How far apart will they be in 1 hour ? After 
traveling 35 hours ? 

19 . There are 100 cents in a dollar. How many cents in 
6 dollars ? 

20. Silver is 10 times as heavy as water. A cubic foot of 
water weighs 62 pounds; what is the weight of a cubic foot 
of silver ? 


M UL TIPLICA TION. 


45 


21. The great Chinese wall has a small tower every 100 
yards. If a man walks beside it, how far has he gone after 
counting 251 spaces between the towers ? 

22. A boy starts on a bicycle from Atlanta, Ha., to Wash¬ 
ington, D. C., a distance of 648 miles. He travels 68 miles 
a day; how far is he from Washington in 10 days ? 

23. Sound travels 1098 feet per second. If you see the 
puff of a steam-whistle 10 seconds before hearing it, how far 
are you from the whistle ? 


39. Copy these examples. Multiply first by-2, by 5, by 8, 
then annex the naught. 


a. 478 

b. 831 

e. 281 

20 

500 

3000 

9460 

415500 

693000 

Examples. 


l. Multiply 478 by 2. 

Multiply the product by 10. 

2. Multiply 881 by 5. 

Multiply the product by 100. 

3. Multiply 281 by 3. 

Multiply the product by 1000. 


4. 144 X 70. ?. 75 X 300. 10. 968 X 6000. 

5 . 1728 X 40. 8. 818 X 4:00, n. 865 X 7000. 

6. 212 X 90. 9. 488 X 500, 12. 492 X 8000. 

13. A bushel of wheat weighs 60 pounds,; how many pounds 
in 212 bushels ? 

14. A bushel of ear-corn (corn on the cob) weighs 70 
pounds; how many pounds in, 528 bushels ? 






46 


THE mth CENTURY ARITHMETIC. 


15. A bushel of bran weighs 20 pounds; how many pounds 
in 75 sacks, each sack containing 2 bushels ? 

16. A degree of a great circle on the earth contains 60 
geographical miles (also called knots). How many such 
miles in 860 degrees; that is, around the earth ? 

17 . In a mile there are 5280 feet; how many feet in 80 
miles ? 

18. The average length of human life is about 80 years. 
How many years are 197 such periods ? 

19. 67 people on the whole earth die each minute, how 
many die in an hour ? 

20. 70 babies on the whole earth are born each minute; 
how many are born in 24 hours ? 

21. How many minutes makes a day ? 

22. How many days in thirty years ? 


40. To multiply by any number. 

Multiply 8617 by 245. 

8617 

245 


5 times the multiplicand = 18085 ) p ,. , 

40 times the multiplicand = 14468 > ar _, ia , 

200 times the multiplicand == 7234 ) P ro uc s ' 

245 times the multiplicand = 886165 

The right hand zeros of the partial products are omitted because they 
do not affect the result of addition. Notice and use as a rule, that 

The right hand figure of each partial product is placed under the 
multiplier used. 4 times 7 = 28; put 8 under 4. 2 times 7 

— 14; put 4 under 2. 





M UL TIPLICA TION. 


47 


Examples. 


a. How much is 31,416 X 5 ? 

b. How much is 31,416 X 6 ? 

c. How much is 31,416 X 3 ? 
(See answers under multiples.) 


31,416 mul¬ 
tiplied by 

1 

2 

3 

4 

5 

6 

7 

8 
9 


Multiples. 

31,416 

62,832 

94,248 

125,664 

157,080 

188,496 

219,912 

251,328 

282,744 


d. 31416 
365 
157080 
188496 
94248 
11466840 


e. 31416 
249 
"282744 

125664 

62832 

"7822584 


Multiply 31,416 by 


1. 

41 

4. 123 

7. 649 

10. 3412 

13. 

6819 

2. 

18 

5. 235 

8. 753 

ll. 3125 

14. 

4765 

3. 

63 

6. 361 

9. 869 

12. 2354 

15. 

7198 


62,622 mul¬ 
tiplied by Multiples. 

1 62,522 

2 = 125,244 

3 = 187,866 

4 = 250,488 

5 = 313,110 

6 = 375,732 

7 = 438,354 

8 = 500,976 

= 563,598 



Multiply 62,622 by 

16. 

41 

26. 

3412 

17. 

18 

27. 

3 25 

18. 

63 

28. 

2354 

19. 

25 

29. 

6819 

20. 

36 

30. 

4765 

21. 

123 

31. 

7198 

22. 

235 

32. 

8543 

23. 

861 

33. 

6119 

24. 

649 

34. 

7984 

25. 

753 

35. 

8197 


9 






48 


THE goth CENTURY ARITHMETIC. 


Multiply 



365 by 


1492 by 


1607 by 


6822 

36. 

23 

46. 

731 

56. 

863 

66. 

177 

37. 

36 

47. 

245 

57. 

491 

67. 

898 

38. 

49 

48. 

176 

58. 

365 

68. 

461 

39. 

18 

49. 

432 

59. 

745 

69. 

365 

40. 

75 

50. 

812 

60. 

913 

70. 

918 

41. 

27 

51. 

473 

61. 

492 

71. 

871 

42. 

68 

52. 

291 

62. 

784 

72. 

643 

43. 

92 

53. 

462 

63. 

314 

73. 

765 

44. 

87 

54. 

365 

64. 

159 

74. 

842 

45. 

95 

55. 

418 

65. 

943 

75. 

684 


Multiply 

76. 528 by 28. 

77. 176 by 86. 

78. 212 by 49. 

79. 492 by 18. 

80. 841 by 21. 

81. 216 by 74. 

82. 876 by 27. 

83. 912 by 38. 


84. 1492 by 731. 

85. 1607 by 245. 

86. 5280 by 176. 

87. 4162 by 432. 

88. 7842 by 812. 

89. 9143 by 473. 

90. 6418 by 291. 

91. 7302 by 462. 


92. 7401 by 831 

93. 1386 by 418. 

94. 4444 by 642, 

95. 6822 by 235 

96. 3466 by 681 

97. 4141 by 263 

98. 2718 by 426 

99 . 3535 by 437 


41. Multiply 4183 by 2005. 

4183 

2005 The partial products by zeros are nothing, and are 
20915 therefore omitted. The first figure of the product by 2 
8366 must be placed under the 2. 

8386915 

Examples. 

Multiply 

1. 1728 by 203. 3. 4761 by 407. 5. 7001 by 405. 

2. 2173 by 107. 4. 4302 by 203. 6. 31,425 by 4001. 





MULTIPLICA TION. 


49 


7. 81,415 by 8002. io. 60,841 by 7004. 

8. 60,728 by 7005. n. 415,846 by 80,008. 

9. 45,302 by 5003. 12 . 613,052 by 40,004. 

13. 714,085 by 50,505. 

14. 323,233 by 30,103. 

15. 430,215 by 50,404. 

16. Georgia has 137 counties with an average area of 424 
square miles in each county. What is the area of Georgia ? 

17. Find the total number of pounds in 35,004 bales of 
cotton if each bale weighs 503 pounds ? 

18. A barrel of flour weighs 196 pounds; how many 
pounds in 404 barrels ? 

19. Find the product of 4005 multiplied by itself ? 

20. A bicycle wheel rotates 705 times in going a mile 
how 'often would it rotate in going from Atlanta to Augusta,, 
Georgia, a distance of 171 miles ? 

21. The president receives $137 per day; how much is that 
per year (365 days) ? 

Multiply 5280 by 1600. 

1st Step. 

5280 Directions.— Put the first integer of the 

1000 multiplier under the first integer of the 
> — multiplicand, naughts falling on the right. 

3168 Multiply by the integer and after adding 

528 partial products, annex the number of 

naughts in both multiplicand and multi¬ 
plier. 

Multiply 

22. 5280 by 2300. 23. by 250. 24. by 1760. 

25. 31400 by 3700. 26. by 360. 27. by 1760. 

28. 63300 by 1500. 29. by 480. 30. by 1760. 


2d Step. 

5280 

1600 

3168 

528 

8448000 






50 


THE goth CENTURY ARITHMETIC , 


42. Numbers that, are multiplied together are called factors. 

(Art. 80). 3 X 5 = 15; 3 and 5 are factors. 

. . . ’ | The dots right and left are called rows, up and 

. down, columns. 

In 1 row there are 5 dots; in 8 rows, 8X5 dots =? 15 dots. 
In 1 column there are 8 dots; in 5 columns; 5 X 8 dots = 
15 dots. 

8 X 5 = 5 X 8. 

43. The product is the same in whatever order the factors are 
arranged. 

This is called the commutative law of multiplication. 

852 X8X5=8X5X 852. a X h = h X a. 


852 352 

8 Notice that the two results are the same. On 15 
1056 the left we multiply 352 by 3 and the product by 1760 

5 5 ; on the right we multiply 352 by 15 (3 X 5 )- 852 

5280 5280 


Examples. 

1. Multiply 728 by 2 and the product by 8. 
This is the same as multiplying by what number ? 

2. Multiply 865 by 15, using the factors of 15. 

3. Multiply 1225 by 21, using factors. 

4. Multiply 1492 by 28, using factors. 

5. Multiply 5280 by 42, using factors. 

6. Multiply 1895 by 85, using factors. 

7. Multiply 149,605 by 86, using factors. 

8. Multiply 365,192 by 21, using factors. 

9. Multiply 405,405 by 24, using factors. 

10. Multiply 840,043 by 56, using factors. 







MULTIPLICATION. 


51 


11. 22,584 X 28. 

12. 81,525 X 27. 

13. 22,584 X 42. 

14. 81,525 X 18. 

15. 22,584 X 27. 


16 . 186,504 x 86. 

17. 186,504 X 21. 

18. 186,504 X 56. 

19. 186,504 X 42. 

20. 186,504 X 28. 


21. 81,416 X 15, 

22. 81,416 X 85 

23. 81,416 X 56, 

24. 31,416 X 63, 

25. 31,416 X 81, 


44. REVIEW. 

1. What is multiplication ? 

2. Name the parts in multiplication. 5x X 3 — 15x. 

3. Define a concrete number; an abstract number. 

4. Explain which parts in multiplication may be concrete 
and which abstract. 

5. When the multiplier is 7 and the multiplicand 9z, 
what is the product ? 

6. What is the sign of multiplication ? Tell the two 
ways of reading this sign and the proper place of the multi¬ 
plier. 

7. The average daily attendance at the World’s Fair was 
149,605. The Fair was open 183 days; what was the total 
attendance ? 

8 . Light travels 186,500 miles in a second and it takes 
493 seconds for the sun’s light to reach the earth; how far 
off is the sun ? 

9. You see a flash of lightning and in 14 seconds hear 
thunder. How far was the lightning, if sound travels 1093 
feet per second ? (The answer is about three miles.) 

10. What is the value of 1728 bales of cotton when the 
average weight of each bale is 473 pounds and cotton is 
worth 7 cents a pound ? 

11. The area of the United States is 3,668,167 square 
miles. In 1890 there was an average 17 people to each square 
mile; what was the total population ? 



52 


THE goth CENTURY ARITHMETIC. 


Each member of the House of the National Congress in 
1897 represented 178,901 people. Find the population of the 
following States: 

Number of Number of 

Representatives. Representatives. 


12 . 

New York, 

84. 

17. Texas, 

18. 

13. 

Pennsylvania, 

28. 

18. Missouri, 

15. 

14. 

Illinois, 

22 . 

19. Georgia, 

11 . 

15. 

Ohio, 

21 . 

20 . Virginia, 

10 . 

16. 

Massachusetts, 

18. 

21 . California, 

7. 


22 . On the whole earth there are 70 births and 67 deaths 
each minute. How much does the population increase in 24 
hours ? 

Diversions.—23. What two numbers multiplied together 
produce seven ? 

24. What is the difference between six dozen dozen and 
half a dozen dozen ? 

25. What is the difference between twice twenty-five and 
twice five and twenty ? 

26. Multiply 10101 by 281. 

Fill out the places marked x in the following with figures: 
(x stands for something unknown.) 


27» 

28. 

29. 

30. 

2x1 

365 

49x 

8 x2 

65 

xl 

x5 

4x 

1155 

365 

2460 

48x2 

1386 

1460 

2952 

32x8 


86892 








SUBTRACTION . 


Multiply 84 by 76. 


84 

76 

6624 

48 

28 

6884 


Notice whence 
the figures are de¬ 
rived gnd where 
the partial prod¬ 
ucts are placed. 


Multiply 

31. 84 by 67. 

32. 96 by 43. 

33. 79 by 87. 

34. 67 by 45. 

35. 89 by 78. 


36. 

75 

by 

75. 

37. 

49 

by 

49. 

38. 

87 

by 

87. 

39. 

95 

by 

95. 

40. 

68 

by 

68. 


In the following examples , what must have been the multiplier f 


1492. 

42. 1492. 

43. 365. 

44. 365. 

X 

X 

X 

X 

5968 

7460 

2555 

3285 

4476 

2984 

1825 

2920 


45. SUBTRACTION. 

// // /. Here are 5 marks. How many must be added 
to make 8 ?■//// / ///. As each new mark is made, say 
the whole number made, (six, seven, eight) till the required 
number are made; then count the number that have been 
added, (one, two, three). By this method you have learned 
that the difference between 8 and 5 is 3. This is the way to 
find the difference between two numbers when the difference 
and number subtracted are both digits. By remembering 
the differences for every pair of digits, there is a process 
called subtraction by which the difference between any two 
numbers may be found. 

The sign of subtraction is — and is called minus. 

8 — 5 = 3 is read 8 minus 5 equals 3. 

After careful consideration of the logic of both methods and exper¬ 
ience in the school-room as to the results of both methods, the mak- 












54 


THE mh CENTURY ARITHMETIC. 


ing-up method for finding differences has been here adopted in 
preference to the taking-away method. One process is as logical as 
as the other; but the making-up method is more nearly related to 
addition and, therefore, to one who knows addition (as does the 
learner at this stage of his progress), it makes a less demand upon 
his conscious attention. The advantage of the method is perhaps 
most apparent in Art. 49, examples 8 to 12. 

If the make-up method is not the superior, why, although taught 
differently, do shopkeepers almost universally use it in making 


change 

1. 

// III 

and how many make 

9? 

5 + 

= 9? 

2. 

II II 

and how many make 

7 ? 

4 + 

= 7? 

3. 

II /I I/ and how many make 

11 ? 

6 + 

= 11 ? 

4. 

// III 

and how many make 

12? 

5 + 

= 12 ? 

5. 

III 

and how many make 

8? 

8 + 

- 8? 


How many added 

6. to // // make 

7. to // III make 

8. to // / make 

9. to // make 

10 . to // // make 


8? 

11 . 

to 

// 

7? 

12. 

to 

// 

9? 

13. 

to 

// 

7 ? 

14. 

to 

// 

6? 

15. 

to 

// 


IIII make 11 ? 

II 11 I make 13 ? 

III make 10 ? 
Illll make 11 ? 
// // make 14 ? 


Point at the following examples and announce the differences: 

The list contains the complete subtraction table, the digits being 
used either as subtrahends or differences. 



a. 

b. 

c. 

d. 

16. 

11 — 6 

5 — 3 

14 — 9 

10 — 7 

17. 

4 — 2 

9 — 7 

11 — 8 

8 — 6 

18. 

18 — 7 

7 — 4 

12 — 6 

18 — 9 

19. 

8 — 4 

10 — 6 

7 — 5 

13 — 9 

20. 

17 — 9 

14 — 8 

13 — 8 

12 — 7 

21. 

6 — 3 

9 — 5 

11 — 7 

11—9 


SUBTRACTION. 


55 


22 . 8 — 5 

23. 17 — 8 

24. 9 — 7 


14 — 7 
6 — 4 
10 — 5 


10 — 8 


9 — 6 
15 — 9 


12 — 8 


16 — 8 


12 — 9 


In the following, speak only the differences; thus, 4, 14, 
24, 84, 44, etc. 

Subtract 

25. 8 from 12, 22, 82, 42, 52, 62, 72. 

26. 5 from 7, 17, 27, 87, 47, 57, 67. 

27. 4 from 9, 19, 29, 39, 49, 59, 69. 

28. 6 from 10, 20, 30, 40, 50, 60, 70. 

29. 2 from 9, 4, 7, 12, 5, 8, 10, 6, 11. 

30. 3 from 5, 8, 13, 6, 9, 11, 7, 12, 10. 

31. 4 from 9, 14, 7, 10, 12, 8, 13, 11, 7. 

32. 5 from 15, 8, 11, 13, 9, 14, 12, 8, 10. 

33. 6 from 9, 12, 14, 10, 15, 13, 9, 11, 8. 

34. 7 from 13, 15, 11, 16, 14, 10, 12, 9, 13. 

35. 8 from 16, 12, 17, 15, 11, 13, 10, 14, 9. 

36. 9 from 13, 18, 16, 12, 14, 11, 15, 10, 13. 


45a. DEFINITIONS. 

Subtraction is the process of finding how many units 
must be added to one number to make a given number. 

The given number is called the minuend. 

The number to be subtracted is called the subtrahend. 
The result is called the difference. 

8—5 = 3. 8 is the minuend; 5 is the subtrahend; and 3 is the 
difference. 

a. Name the parts in 12 — 7 —5; in 17 — 9 = 8. 

b. Subtrahend =5 ; minuend = 11; what is the difference? 
C. Minuend @ 15; subtrahend = 7; what is the difference? 



56 


THE noth CENTURY ARITHMETIC. 


Only like numbers can be subtracted. 

46 . A man having $4769, spent $1525. How much had 
he left ? 

$4769 

1525 Directions. —Write the numbers with the same orders 

$3211 under each other. Begin on the right and subtract. 


1 . 

2. 

3. 

4. 

5. 

Dollars. 

Bushels. 

Pounds. 

Miles. 

Gallons. 

763 

8476 

45,362 

84,276 

98,789 

421 

4371 

42,132 

43,241 

44,332 


6. From 896 dollars subtract 493 dollars. 

7. From 7308 take 4105. 

8 . From Atlanta to San Francisco is 2,847 miles. After 
traveling from Atlanta 1,523 miles, how far are you from 
San Francisco ? 

9. The minuend is 85,613; the subtrahend 43,401; what 
is the difference ? 

10. 956,814 — 433,612 = ? n. 986,487 — 743,255=? 


12. 13. 

864,942 376,518 

332,741 255,313 


47 . A man having $5281, 
he left ? 


14. 15. 

684,296 , 743,891 

284,173 621,531 


spent $1728; how much had 


Th. H. T. U. 
$ 5 2 8 4 
17 2 8 

$ 3 5 5 6 


Directions. —Write the numbers with the 
same orders under each other. Begin on the right 
to subtract. Say: 8 and 6 (write 6) are 14; 3 
and 5 (write 5) are 8; 7 and 5 are 12; 2 and 3 
are 5. 


Explanation. —We add and subtract by remembering the sums of 
every pair of digits. Now the sum of two digits can not be greater 
than 18, (9 + 9). Therefore, the digit which was added to 8 must 














SUBTRACTION. 


57 


have been one which made the sum 14. In adding, we say 8 and 6 are 
14, write 4 and carry 1; then we say 8 and 5 are 8. So, in subtracting 
we carry the 1 from 14 and say 8 and how many make 8; etc. 


Dollars. 

885 

472 


Bushels, 

4872 

2547 


Pounds 

7532 

4265 


Miles. 

84,273 

56,647 


71,543 x 
46,271 x 


1. The distance through the earth at the poles is 7899 
miles; at the equator, 7925 miles. What is the difference ? 

2. How many years has it been since Columbus discov¬ 
ered America in 1492 ? Since the declaration of independ¬ 
ence in 1776 ? Since Washington was make president in 
1789? 

3 . From New York to Liverpool is 3040 miles, and to 
San Francisco is 3269 miles; what is the difference ? 


Find the increase in population 
1890. 

4. Georgia, 1,837,353; 

5. Texas, 2,235,523; 

6. New York, 5,997,853; 

7. Pennsylvania, 5,258,014; 


of the following States: 
1880. 


1,542,180. 

1,591,749. 

5,087,871. 

4,282,891. 


8. The United States: in 1890, 62,622,250; in 1880, 
50,155,783. 

9 . The area of the globe is 196,096,060 square miles, of 
which 143,621,181 square miles is water. How much land 
is there ? 

10. The population of England in 1891 was 27,482,104, 
and of Great Britain 37,888,153. What was the difference? 

11 . From 1,000,000 take 314,159. 

12. In 1892 Georgia raised 1,100,000 bales of cotton; in 
1893, 830,450 bales. What was the difference ? 








58 


THE 20th CENTURY ARITHMETIC. 


13. The average yearly production of wheat in the United 
States is 458,077,791 bushels. How much is that less than 
five hundred million? 

14. In 1890 the number of children of school age in the 
United States was 18,799,864, and the number of actual 
pupils was 8,829,284. How many children were not attend¬ 
ing school ? 

15. When the subtrahend is 62,622,250 and minuend a 
hundred million, what is the difference ? 

16. The careers of 12,483 young men were watched. In 
10 years, whiskey had killed 634, smoking cigarettes killed 
405, and 841 died of other troubles. How many were then 
living ? 

17. From 987,654,321 subtract 123,456,789. 

18. To what number must 142,857 be added to make 
285,714 ? 

19. From 100,101,001 subtract 62,622,250. 

20. What is the difference between the largest and small¬ 
est cotton crop mentioned on page 31 ? 

For other examples seepages 31 and 32. Use the total population 
as minuend and number of either sex as subtrahend. 


48. REVIEW. 

1 . What is subtraction ? 

2. Name the parts in subtraction: 5 — 3 = 2. 

3. What is the sign of subtraction and its name ? 

4. When the minuend is 10, the subtrahend 7, what is 
the difference ? 

5. The distance around the earth is 24,899 miles; the 
distance through it is 7925. Subtract the latter from th e 



ADDITION AND SUBTRACTION. 


59 

former and the successive remainders as many times as you 
can. 

6. 23,428,423 — 9,056,273 •-= what ? 

7. Use 5280 and successive remainders as minuends and 
1760 as a subtrahend 3 times. 

8. The attendance at the World’s Fair in 1893 was 
27,377,733; at the Paris Exposition in 1889, 32,354,111. 
What was the difference ? 

9. What number must be added to 7,965,499 to make 
13,642,809 ? 

10. From the difference between 3285 and 456 subtract 
the difference between 19,011 and 17,455. 

11. From the largest number that can be expressed by the 
figures 7, 2, 6, 3, and 5, take the smallest number that can 
be expressed by the same figures. 

12. The sum of two numbers is 917; one is the difference 
between 693 and 1201; find the other. 


49. ADDITION AND SUBTRACTION. 

1. Three addends give the sum 864; two of the addends 
are 173 and 445, what is the other ? 

2. The minuend is 243; the sum of 125 and the subtra¬ 
hend is 271; what is the difference ? 

3. The sum of the minuend and subtrahend is 763; the 
minuend is 527; what is the subtrahend and what the dif¬ 
ference ? 

4. Two separate examples in subtraction have equal sub¬ 
trahends; the minuends are 643 and 428, and the sum of the 
differences is 823; what is the subtrahend of each? 

5. The minuend is 22,222 greater than the subtrahend; 
the subtrahend equals the remainder; the remainder is 
37,564; what is the minuend ? 



60 


THE mth CENTURY ARITHMETIC. 


6. The population of a city on January 1, 1897, was 94,748. 
During the year there were 2875 births, 1846 deaths, 947 
people moved in, and 459 moved away. What was the pop¬ 
ulation January 1, 1898 ? 

7. Two boys rode on bicycles from Atlanta in opposite 
directions: 


◄- Roy. A. Walter. -► 


Days 

8d. , 

2d 

i 1st. 

1st. 

, 2d. 

, 3d. 

Miles 

68 ‘ 

65 

1 71 | 

68 1 

1 78 

1 59. 


At the end of the third day how far had each traveled, 
and how far were they apart ? 

Subtract the sum of 1492, 1776, and 865 from 5280. 

Directions. —Place the numbers and draw the lines as 
shown. Beginning on the right, add by columns up to the 
upper line, then ask yourself how many more must be 
added to make the number above the line. In this ex¬ 
ample, say, adding first column : 5, 11, 13, and 7 make 20, 
(write 7 and carry 2). Adding second column: 8, 15, 24, 
and 4 make 28, (write 4 and carry 2). Adding third col¬ 
umn : 5,12, 16, and 6 make 22, (write 6 and carry 2). Add¬ 
ing fourth column : 3, 4, and 1 make 5, (write 1). 

In the following examples, add the numbers between the lines 
and subtract the sum from the number above the top line: 


5280 

1492 

1776 

865 

1647 


8. 

9. 

10. 

n. 

12. 

12765 

6004 

39370 

10,000 

61,027 

3872 

1776 

5760 

31,416 

16,093 

5280 

1492 

365 

31,416 

16,093 

1760 

1898 

3937 

31,416 

16,093 


13! The crown used by the English sovereign contains 8,091 
jewels. Of these, 1861 are brilliant diamonds; 1278 are 
rose diamonds, and 147 are table diamonds. How many 
other jewels are in the crown ? 














A DDITTON A ND S UB TRA C TION. 


61 


14, Length of rivers: Amazon, 8944; Nile, 8800; Missis¬ 
sippi, 2616. The sum of these three lengths and that of the 
Missouri, is 18,456. How long is the Missouri river ? 


50. Diversions.— Take a set of numbers which increase in 
value. Subtract each from the one next greater. The sum 
of the remainders and smallest number should equal the 
largest number. 


1. Numbers. 

Remainders. 

2. Numbers. 

3. Numbers. 

275895 

207465 

864293 

361420 

68430 

66287 

215642 

28419 

2143 

1947 

38291 

19142 

196 


6425 

3562 


Make up similar examples. 

4. Take any number, say 365,847. Subtract 2 365,847 

andprefix2: 2,365,845; this will be the sum. Take 258,241 
any other number, placing it under the first one 741,758 
for addition. Under the second number write a 429,580 
third so that the sum of the digits in the same orders 570,419 
of the 2d and 3d numbers shall be 9 (see ex- 2,365,847 
ample). Take any fourth number and write a 
fifth under it according to the directions in italics. The sum 
of the five numbers will be as stated. 

Make up similar examples. 

5. 314,159 6. 149,265 7. 17,761,607 

261,348 810,990 52,801,444 

614,280 946,000 15,651,583 


2 , 314,157 


2 , 149,263 


217 , 761,605 







62 


THE mh CENTURY ARITHMETIC. 


51. DIVISION. 


///////////////. Here are 15 marks. Separate them 
into groups of three marks each and count the number 
of groups. HI HI I// I// HI; 5 groups. This is the 
way to find a quotient (5) when it and the divisor (3) are 
both digits; but by remembering all such quotients, there 
is a process called division by which the quotient of any two 
numbers may be found. 

The sign of division is -r-. 

15 -r- 3 = 5 is read 15 divided by 3 equals 5. 

3 and 5 are how many ? Addition. 3 and how many are 8 ? Sub¬ 
traction. 3 times 5 are how many ? Multiplication. 3 times how 
many are 15 ? Division. 


52. How many groups of 

1. // make 12 marks ? 

2. I// make 15 marks ? 

3. 11 /1 make 16 marks ? 

4. 11111 make 15 marks ? 

5. 11111 make 20 marks ? 
The answer to each of these ten 


6. Ill make 21 marks ? 

7. /Ill make 24 marks ? 

8. ///// make 30 marks ? 

9. I// make 30 marks ? 

10. //// make 28 marks ? 

questions is called a quotient. 


How many groups of //// 

mi //// 

How many groups of 

11. Ill make 13 marks? 

12. Ill make 17 marks? 

13. Jill make 22 marks ? 

14. 1111 make 14 marks ? 

15. ///// make 17 marks? 


make 14 marks ? 

Three groups and two marks. 

16. //// make 22 marks ? 

17. 1111 make 35 marks ? 

18. I// make 28 marks ? 

19. I// make 17 marks ? 

20 • 11111 make 34 marks ? 

the number of 


The number of groups is called the quotient; 
marks left is called the remainder. 


63 


DIVISION. 

21 . 3 times what number is 15 ? 

22. 3 times what number is 27 ? 

23. 4 times what number is 20 ? 

24. 4 times what number is 28 ? 

25. 5 times what number is 30 ? 

26. 6 times what number is 42 ? 

27. 7 times what number is 63 ? 

28. 8 times what number is 56 ? 

29. 9 times what number is 72 ? 

30. 9 times what number is 54 ? 


15 -s- 3 = 

27 -4- 3 = 
20 -4- 4 = 

28 -s- 4 — 

30 -r- 5 = 
42 ~ 6 = 
63 h- 7 = 
56 -i- 8 = 
72 9 — 

54 9 = 


53. Point at the following examples and give the quotients: 

The list contains the complete division table, the digits being used 
either as divisors or as quotients. To give the quotient in 11 a, the 
pupil should ask himself, “6 times what number is 18?” and similarly 
for the others. 



a. 

b. 

c. 


d. 

1 . 

18-1-6 

9-i-3 

36 -- 

9 

42-1-7 

2. 

4-2 

81-1-9 

16 - 4 - 

8 

36-4-6 

3. 

35 -4- 7 

12-1-4 

24 - 4 - 

6 

45-1-9 

4. 

16-4-4 

12-4-6 

15 - 4 - 

5 

72-1-9 

5 

10 - 4 - 5 

40-4-8 

32 - 4 - 

8 

49-1-7 

6 

63-1-9 

25-1-5 

14 - 4 - 

7 

54-4-9 

7. 

6-4-3 

28-4-7 

30 - 4 - 

6 

48-1-8 

8. 

56-1-8 

8-4-4 

18 - 44 - 

9 

27-1-9 

9. 

21-4-7 

20 - 4 - 5 

24 -i- 

8 

oo 

o 

number is not divided by 1. 

The operation may be expressed 


but the result shows that no division has taken place. 


54 . DEFINITIONS. 

Division is a process for finding by what a number must 
be multiplied to produce a given number. 







64 


THE 20th CENTURY ARITHMETIC. 


The parts in division are called divisor, dividend, quo¬ 
tient, and remainder. 

17 -4- 3 = 5 and remainder 2. Here 17 is the dividend, 3 
the divisor, 5 the quotient, and 2 the remainder. 

Name the parts in 21 4- 3 = 7; in 63 7 = 9. 

55. Find what must have been the multiplier in the following 
examples: 


1 . 

2. 

3. 

4. 

5. 

231 

432 

233 

412 

408 

693 

864 

699 

1648 

2448 

6. 

7. 

8. 

9. 

10. 

315 

412 

913 

742 

365 

1575 

2472 

2739 

5936 

2555 


56. SHORT DIVISION. 

Divide 693 by 3. (See Art. 36.) 

Directions. —Draw a line about the dividend as shown. 
231 Divide each digit of the dividend by the divisor. By this 
qfnqq process you have found that 3 must be multiplied by 231 to 
I produce 693; or that 693 lines will make 231 groups of 3 
lines each. 


Examples. 

1. 864-4- 2, 6. 963 4 - 3. n. 884 4 - 4 . 16. 1505 -s-5. 

2. 428 -4- 2. 7. 663 -4- 3. 12. 848 4-4. 17. 1806 4 - 6. 

3. 644 - 4 - 2. 8. 996 4 - 3. 13. 1284 - 4 - 4. is. 2807 - 4 - 7. 

4. 482 -4 2. 9. 366 4 - 3. 14. 1648 4 - 4. 19. 1616 4 - 8. 

5. 866 -4 2. 10. 699 4- 3. 15. 2084 -4- 4. 20. 2727 — 9. 

57. Read Articles 23 and 28, II. 

One unit of any order makes ten units of the next lower order. 
A figure of any order may be read as tens of the next lower order . 









SHORT DIVISION. 


65 


Divide 5184 by 8. 

Directions. —Place the numbers and draw 
1 < 2 8 the line as shown. 3 is contained in 5 once and 

Th. H. T. U. 2 over. This remainder 2 has not yet been di- 
5 18 4 vided: by reduction, it becomes 20 hundreds, 

and the 1 in hundreds place gives 21 hundreds; 
which divided by 3 gives 7. Thus each remainder is simply prefixed 
to the next right hand order (which makes it tens of that order). 3 is 
contained in 8 twice and 2 over; and into 24, 8 times. Copy this ex¬ 
ample. 


Examples. 


Divide 

1. 730 by 2. 

2. 483 by 3. 

3. 624 by 4. 

4. 852 by 4. 

5. 786 by 3. 
58 . Divide 4378 


6. 2555 by 7. 

7. 31,680 by 6 

8. 1696 by 8. 

9. 2996 by 7. 
10. 4476 by 3. 
4. 


11. 9905 by 7. 

12. 325,938 by 6. 

13. 69,055 by 7. 

14. 4347 by 9. 

15. 7460 by 5. 


1094 


When the divisor is not contained in the partial divi- 
^ 9 dend it is so indicated by naught in the quotient. 

41To to When there is a remainder, it should be written as 

is 2 in the example. This shows that if two were subtracted from 
the given number the exact quotient would be 1094. 


1. 4376 -4- 4. 

2. 1826 4- 3. 

3. 20,518 -4- 5. 

4. 12,411 - 4 - 2. 

5. 46,808 -4- 6. 


6. 24,026 4 - 6 

7. 14,087 4- 7 

8. 8749 4 - 8. 

9. 6343 4- 9. 
10. 1221 4 - 4. 


11. 37,869 4 - 9. 

12. 26,404 4 - 5. 

13. 9843 4 - 9. 

14. 11,260 -4 7. 

15. 8749 4 - 8. 


16. Divide 5280 feet (one mile) by 3 feet. 3 feet make 
one yard; what is your quotient ? 

17. I heard the sound of a whistle 7 seconds after seeing 
the puff of steam, and, measuring the distance, found I was 







66 


THE 20th CENTURY ARITHMETIC. 


7651 feet from the whistle. How far does sound go in 1 second? 

18. Around a circle is 3 times across it. What is the dis¬ 
tance, in feet, across a circle w r hich is 1 mile (5280 feet) 
around ? 

19. How many steps of 4 feet each would a man take in 
running 5 miles ? 1 mile = 5280 feet. 

20. If a bicyclist goes 9 miles an hour, how many hours 
will he be in going from Boston to Atlanta, a distance of 
1089 miles? 

21. A bicycle wheel is 7 feet around. How often will it 
rotate in going 1 mile (5280 feet)? 

22. Iron weighs 8 times as much as wafer. A certain 
quantity of iron weighs 8744 pounds; what does the same 
quantity of water weigh ? 

23. In 7 square miles there are 4480 acres; how many 
acres in 1 square mile ? 

24. 9 cubic feet of water weigh 738 pounds; what does 1 
cubic foot of water weigh ? 

25. How many 5 grain capsules can be filled with 5760 
grains (1 pound) of quinine? 

26. A boy said: If I had 3 times my number of marbles 
and 5 more, I should have 185. How many have I ? 

27. A dog is 225 feet behind a rabbit. The rabbit runs 
16 and the dog 21 feet a second. How many seconds before 
the dog will catch the hare ? 

28. A boy bought papers at 4 for 8 cents and sold them 
at 5 cents apiece. How many papers had he sold when his 
gain was 525 cents ? 

29. 9 cubic feet of water weigh 558 pounds; what is the 
weight of 1 cubic foot ? 

30. If a bicyclist rides 455 miles in 7 days, how far does 
he ride in a day ? in 5 days ? 


SHORT DIVISION. 


67 


31. If the Mississippi flows 3 miles per hour, how many 
hours does it take for water from its source, Lake Itasca, to 
reach the Gulf of Mexico ? The Mississippi is 3162 miles 
long. 

32. Mr. King paid $9500 for a farm, and sold it for 
$8000, losing $3 per acre; how many acres were in the farm? 

59. The sign of division is but division may also be 
indicated by writing the divisor under the dividend with a 
line between. 


12 -f- 3 = Jj 2 - = 12 divided by 3. 
In this way indicate the following divisions: 


a. 

1. 12 3. 

2. 15 5. 

3. 18 -r- 2. 

4. 17 6. 

5. 14 -v- 3. 


b. 

24 -f- 7. 
19 11. 

27 M 15. 
35 8. 

18 hh 13. 


c. 

4 -f- 9. 

7 13. 

1 - 4 - 3. 

2 - 4 - 5. 

14 23. 


d. 

17 83. 

14 -4- 97. 
73 141. 

62 -f- 187. 
49 301. 


60 Division consists in dividing the dividend and then each 
successive remainder by the divisor. The last remainder is 
always smaller than the divisor and therefore cannot be 
actually divided, but the division should always be indicated 
as above. 

¥ = 4§. if = 1 T V HP- = 67*. 

Remainders written in this way are called fractions. 
(Art. 106.) The figure above the line is called the nume¬ 
rator; the figure below the line is called the denominator. 
When the denominator is small, a fraction is read by read¬ 
ing its numerator as a cardinal number and its denomina¬ 
tor as an ordinal number. 

Cardinal numbers are one, two, three, four, etc. Ordinal 
numbers are, first, second, third, fourth, etc. 

§ is read two-thirds. | is read seven-ninths. 


68 


THE 20th CENTURY ARITHMETIC. 


Read: 

2 

71 
7 

^ is read one-half instead or one-second. 

61 . When the denominator is large, a fraction is usually 
read as an expression of division. is read, 27 divided 

by 145. 


TT 

I 


Read: 


1 2 3 
STT 


¥ — 4f is read, 14 thirds equals 4 and § . 


2_3_2-8 _ e 7 V T . The quotient is read, 8 and 45 divided by 


76 1 

761. 


Write the quotients to the following: 


i. ¥• 

6. V-. 

ii. 

2. -V-- 

7 Ap- 

■ 12. 

3. ¥• 

q 6 8 5 

«. -jj-. 

13. 

4. J-P. 

9. 

14. P 6 - 0 , 

5. ¥ A - 

10. J - 6 4 0 --- 

15. P/A 


62. LONG- DIVISION. 

Divide 14,880 by 32. 

465 Directions. —Place the numbers and draw the lines 

32 1 14880 as shown. Find each figure of the quotient as in short 
128 division, using only the left figure of the divisor. 
'208 Thus, 3 is contained in 14, 4 times ; multiply the whole 
192 divisor, 32, by this figure of the quotient, subtract, 
160 and bring down the next fi S ure > 8 - 8 is contained in 
160 20,6 times > multiply 32 by 6 , subtract, draw down 

- next figure, and so on. 







LONG DIVISION. 


69 


Place each digit of the quotient over the right hand digit 
of the dividend used in obtaining it. 

Copy this example and study the directions. 

Explanation. 


32 X 400 = 
32 X 60 = 
32 X 5 = 

12800 

1920 

160 

Obtain from this ) 
the products j 

32 
32 
' 32 

X 65, 

X 460, 
X 405. 

32 X 465 = 

14880 • 

32 multiplied by 

Divide by 

32: 


2 = 64. 

6 - 192. 

1. 800. 

6. 

11,680. 

3 = 96. 

7 = 224. 

2. 1024. 

7. 

13,600. 

4 = 128. 

8 = 256. 

3. 1312. 

8. 

13,504. 

5 = 160. 

9 = 288. 

4. 2336. 

9. 

17,744. 

Divide 


5. 2684. 

10. 

31,296. 


11 . 

14,880 

by 32. 

19. 

29,575 

by 

91. 

27. 

20,632 

by 

72. 

12. 

11,680 

by 32. 

20. 

11,985 

by 

51. 

28. 

22,608 

by 

72. 

13. 

800 by 

32. 

21. 

18,615 

by 

51. 

29. 

66,584 

by 

82. 

14. 

2336 by 32. 

22. 

42,903 

by 

63. 

30. 

28,290 

by 

82. 

15. 

13,600 

by 32. 

23. 

27,405 

by 

63. 

31. 

24,722 

by 

94. 

16, 

9963 by 41. 

24. 

14,175 

by 

75. 

32. 

35,814 

by 

94. 

17. 

40,768 

by 52. 

25. 

39,600 

by 

75. 

33. 

62,220 

by 

85. 

18. 

27,531 

by 63. 

26. 

26,062 

by 

83. 

34. 

20,825 

by 

85. 


63 . Divide 1541 by 23. 

67 Directions. —Follow directions in Art. 62. Using the 

oorpnT left figure of the divisor as before, we say, 2 is con- 
' f tained in 15, 7 times ; but multiplying 23 by 7 gives 161, 

138 which is more than 154. Therefore 23 is not contained 

161 in 154 as many as 7 times. Try the next lower figure, 6, 
^g^ and so on. 

- Copy this example till you can perform the division per¬ 
fectly.' 

From this example, you see that the left figure of the divisor is 
simply used as a trial divisor. There is no other way to find the true 
figure of the quotient than to make these trials. 







70 


THE goth CENTURY ARITHMETIC. 


Examples. 


1. 

1426 -4 

- 23. 

n. 

109,935 - 

- 35. 

21. 

14,322 - 

- 231. 

2. 

736 4 

- 23. 

12. 

9855 

-27. 

22. 

7200 - 

- 225. 

3. 

1288 - 

4 23. 

13. 

85,248 - 

- 48. 

23. 

24,192 - 

- 432. 

4. 

5313 - 

4 23. 

14. 

96,980 - 

- 65. 

24. 

53,361 - 

- 231. 

5. 

3312 - 

4 23. 

15. 

17,808 - 

- 84. 

25. 

133,125 - 

- 365. 

6. 

8642 - 

4 58. 

16. 

18,212 - 

-58. 

26. 

165,792 - 

- 528. 

7. 

7807 - 

4 37. 

17. 

30,624 - 

- 58. • 

27. 

92,928 - 

- 176. 

8. 

4847 - 

4 37. 

18. 

21,170 - 

- 58. 

28. 

133,225 - 

- 365. 

9. 

5513 4 

4 37. 

19. 

10,256 - 

-58. 

29. 

435,600 - 

- 528. 

10 . 

6549 4 

4 37. 

20 . 

41,412 - 

- 58. 

30 . 

118,096 - 

- 176. 


64. When the divisor is not contained in the partial 
dividend, it is so indicated by naught in the quotient. In¬ 
dicate the remainders as explained in Art. 60. 


Execute these divisions: 

1 . 1098 ^f^ 

8651899072 

3 . 281,747 •+■ 212. 

4 . 451,401 - 4 - 144. 

5 . 334,800 - 4 - 314. 

6. 231,819 -r- 1093 

7 . 371,943 - 4 - 1607, 


2. 5280 t Vj 

144| 760417 

8. 2,102,423 - 4 - 365. 

9 . 371,300 - 4 - 231. 

10. 335,000 - 4 - 1066. 

11. 451,391 - 4 - 3141. 

12. 2,102,573 -4- 5760. 


65. To. Divide by 10, 100, Etc. 

To multiply a number by 10, each digit is moved one place 
to the left , (Art. 38); conversely, to divide by 10, each digit 
should be moved one place to the right. 

Remove the digits of 1492 one place to the right. 

This is done by putting a period between 9 and 2 to show 
that the integral part of the number ends with 9. 







LONG DIVISION. 


71 


Th. H. T. U. Th. H. T. U. 

1 492-- 10 = 149.2 

To divide by 100, or 10 X 10, point off two places. 

To divide by 1000, or 10 X 10 X 10, point off three 
places, etc. 

1492 -- 100 = 14.92. 1492 = 1000 = 1.492. 


The integral (Art. 77 d.) 
—lA = 14.92 quotient is the number to 
100 the left of the period and the 

— 14 x 9 q 2 o number on the right is the 
remainder. 


1192. 

^100 

10011492 
100 
492 
400 
92 


1. 1492 = 100. 

2. 1728 100. 

3. 1092 h- 100. 

4. 5280 -r- 100. 

5. 8141 v- 100. 


8. 492 10. 

7. 145 -l- 10. 

8. 865 -- 10. 

9. 848 10. 

10. 921 h- 10. 


11. 1092 = 1000 

12. 5280 h- 1000 

13. 8141 1000 

14. 6721 = 1000 

15. 4878 -4- 1000 


66. Since 500 = 5 X 100; 170 =17 X 10; etc., we have 
Rule: To divide by a number ending in naughts , point off 
the naughts in the divisor and the same number of places in the 
dividend before dividing. 


Oil 6 49 
" II 2500 

25.00|5281.49 


The remainder obtained by 
using the divisor without 
naughts should be prefixed to 
the figure cut off from the 
dividend. The reason is ap¬ 
parent from the example on 
the right. 


911 649 

25.00|5281.49 

50 

28 

25 

31 

25 


649 














72 


THE mh CENTURY ARITHMETIC. 


1. 

6248 

-4- 40. 

ll. 

31,685 -f- 

600. 

21 . 

314,159 -4- 

3000. 

2 . 

7869 

-4- 30. 

12 . 

32,593 4- 

500. 

22 . 

784,291 -4- 

6000. 

3 . 

2555 

-4- 70 

13 . 

69,055 -r- 

200. 

23 . 

477,125 + 

2500 

4 . 

4347 

-4- 90 

14 . 

48,347 -4- 

800. 

24 . 

648,192 -j- 

3200. 

5 . 

7462 

-4- 50. 

15 . 

25,004 -4- 

400. 

25 . 

829,146 -4- 

2100. 

6 . 

3146 

~r- 20. 

16 . 

31,416 -4— 

500. 

26 . 

271,828 -f- 

2300. 

7 . 

5280 

30. 

17 . 

86,191 

700. 

27 . 

814,159 - 

4500. 

8 . 

3937 

-4- 20. 

18 . 

24,222 h- 

300. 

28 . 

141,421 -4- 

1600. 

9 . 

7462 

-4- 50. 

19 . 

36,524 hh 

600. 

29 . 

154,323 -4- 

3700. 

. 10 . 

3146 

-4- 60. 

20 . 

39,371 -f- 

900. 

30 . 

223,606 -4- 

5900. 


67 . Averages. 


365 

The average of several numbers 

609 

416 

is a number which will produce the 

609 

984 

same sum if put in the place of each 

609 

671 

of the given numbers. It is found 

609 

2436 

by dividing the sum by the number 
of numbers. The average of the 
four numbers on the left is 609. 

2436 


2486 -s- 4 = 609. 

What is the average of 

1. 5, 6, and 9. 4 . 6, 7, 8, and 11. 

2. 2, 6, and 7. 5 . 5, 7, 10, and 14. 

3. 8, 4, and 8. 6. 7, 9, 11, and 18. 

7. 5 bales of cotton weigh respectively 480, 503, 491, 475, 
and 511 pounds. What is the average weight ? 

8. Several bushels of wheat weighed respectively 61, 59, 
58, 60, 59, and 63 pounds. What was the average weight ? 





LONG DIVISION. 


7 6 


9 . The daily attendance at Ivy Street School for a 
week was 425, 432, 429, 437, and 432. What was the average 
daily attendance ? 

10 . 6 careful experiments were made to find the velocity 
of sound; the results in feet per second were 1093, 1096, 
1104, 1092, 1087, and 1090. What was the average ? 

11 . John and Henry are each 15 years old; Mary is 14, 
and Roy and Sam each 13. What is the average age of these 
five ? 

12. In a certain class two of the pupils are 10 years old; 
six are 11; twelve are 12; and ten are 13. What is the average 
age of all these pupils ? 

13. What is the average of 6, 7, 8, 9, and 10 ? What is 
the average of 6 and 10 ? 

14. What is the average of 5, 7, 9, 11, and 13? What is 
the average of 5 and 13 ? 

What is the average of the following: 

15. 4, 7, 10, 13, and 16. Of 4 and 16 ? 1 

16. 1, 5, 9, 18, and 17. Of 1 and 17 ? I Notice the dif- 

17. 2, 7, 12, 17, and 22. Of 2 and 22 ? ferencer 

18 . 5, 11, 17, 22, and 27. Of 5 and 27 ? , 

19 . Count by 2’s from 3 to 15. Find the sum of these 

numbers, including 3 and 15. 

20. Count by 3’s from 5 to 23. Find the sum of these 
numbers, including 5 and 23. 


68. REVIEW. 

1. What is division ? 

2. Name the parts in 17 divided by 3 equals 5 and 2 over. 

3 . Explain which parts may be concrete and which ab¬ 
stract numbers. 




74 


THE goth CENTURY ARITHMETIC. 


4. Define Short and Long divisions. 

5. What is the trial divisor ? 

6. Show the two ways of indicating division. 

7. What is an average ? 

8. Which parts are the same in multiplication and divi¬ 
sion ? Div. 15a: -4- 8a: = 5. Mul. 3a; X 5 = 15a;. 

9. If the divisor and quotient are given, how do you find 
the dividend ? 

10. If the dividend and quotient are given, how do you 
find the divisor ? 

11 . If the divisor, quotient, and remainder are given, how 
do you find the dividend ? 

12. The hot and cold water faucets of a bath tub which 
holds 128 gallons are turned on. 21 gallons of hot water 
flow in 3 minutes and 63 gallons of cold water in 7 minutes. 
How long before the tub is half full ? 

13 . A ship travels 23 miles per 'hour. How many hours 
will it take to go from New York to Liverpool, 3040 miles ? 

14. A train averages 25 miles in going from New York to 
San Francisco, 3269 miles. How many hours does it take 
to make the trip ? 

15. The area of Georgia is 58,000 square miles and the pop¬ 
ulation in 1890 was 1,837,353. What was the average num¬ 
ber of people per square mile ? 

16. New York. Area 47,000 square miles; population in 
1890, 5,997,853. Average population per square mile ? 

17. The President of the United States receives $50,000 in 
365 days (1 year). How much is that per day ? 

18. It is 25,000 miles around the earth. How many miles 
per day must one travel to go round in 80 days ? 

19. The World’s Fair was open 183 days; the total attend¬ 
ance was 27,377,733. What was the average per day ? 


LONG DIVISION. 


75 


20. A typewriter writes 25 pages in 45 minutes. Each 
page has 25 lines with an average of 10 words to the line. How 
many words can she write per minute ? 

21 . There are 16 ounces in a pound. How many pounds 
in 4625 ounces ? 

22. How many square miles in Georgia ? There are 
37,120,000 acres, and 640 acres make one square mile. 

23 . In 1893 Georgia raised 1,732,824 bushels of wheat on 
15,200 acres. What was the average yield per acre ? 

24 . In 1893 Georgia raised 33,678,000 bushels of corn on 
149,680 acres. What was the average yield per acre ? 

25 . The earth turns on its axes 360 degrees in 24 hours. 
How many degrees does it turn in 1 hour ? 

26 . If gold is worth $240 per pound, how many pounds of 
gold are in one million dollars ? 

27 . What is the nearest number to 11,575 that will con¬ 
tain 221 without a remainder ? 

28 . How many bushels of corn will weigh as much as 224 
bushels of wheat ? A bushel of wheat weighs 60 pounds; of 
corn 56 pounds. 

29 . If 16 gold dollars weigh as much as one silver dollar, 
how many silver dollars weigh as much as 2480 gold dollars? 

30 . A cubic foot of gold weighs 1203 pounds, and is worth 
$362,380; what is one pound of gold worth ? 

31. In the Civil War the entire Union force was 2,653,062 
men. If 1 for every 65 was killed, and 1 for every 10 
wounded, how many were killed and how many were 
wounded ? 

32 . 10 eggs weigh one pound. How many pounds will 15 
dozen eggs weigh ? 

33. Diversion.— A piece of cloth is 50 yards long. If you 
cut off two yards every day, how many days will it take to 
cut it into two-yard pieces ? 


76 


THE 20th CENTURY ARITHMETIC. 


34. 231 cubic inches make a gallon. How many gallons 
in 2,333,331 cubic inches ? 

35. A certain number is multiplied by 95 and 225 is sub¬ 
tracted from the product; the remainder divided by 125 
gives a quotient 37. What is the number ? 

Suggestion. 125 | Remainder Bematader + 225 = product. 

It requires 173,901 people for each representative in the 
House of Representatives of our national Congress. To how 
many representatives is each of the following States entitled 
and how many people are left unrepresented ? 


Population. 

36. New York, 5,997,853. 

37. Pennsylvania, 5,258,014. 

38. Illinois, 3,826,351. 

39. Ohio, 3,672,316. 

40. Massachusetts, 2,238,943. 


Population. 

41. Texas, 2,235,523. 

42. Missouri, 2,679,184. 

43. G-eorgia, 1,837,353. 

44. Virginia, 1,655,980. 

45. California, 1,208,130. 


69. Signs of Aggregation. 

Parentheses ( ), are used to enclose numbers upon 

which one operation is to be performed. 7 + ( ) means 

that the quantity in parentheses is to be added to 7. 

7 — ( •), the quantity is to be subtracted from 7. 

( ) X 7, the quantity is to be multiplied by 7. 

( ) -r- 7, the quantity is to be divided by 7. 

The vinculum , brackets [ ], and braces ( ) 

are used like parentheses. } ) 





SIGNS OF AGGREGATION. 


77 


They are called sign of aggregation. 

7 + (8 — 3) = 7 + 8^8 = 7+ [8 — 3] = 7 + | 8 — 8 j 

Rule .—Find the value of the expression in parentheses before 
performing the indicated operation. 

7 + (8 — 8) . Find the value of 8 — 3 before adding. 

8 — 3 = 5; 7 + 5= 12. 

7 — 8^3 = 7 — 5 = 2. [8 —3] X 7 = 5 X 7 = 35. 

Multiplying and dividing should be performed before 
adding or subtracting. 

1. 9 + (11 — 4). 9. 56 7 -f 12 X 8 — 52 = 2. 

2. 9—(11—4). io. (491 — 275 —”95) X (45 — 16). 

3. (11 — 4) X 9. li. 81 — (108 = 9) + 43. 

4 . (67 —4)^9. 12 . 148—15 + 31> 6. 

5 < (7 _|_ 4) — (11 — 5). When multiplication and divi- 
_ sion signs occur they should be 

6. 3 + 4 — 2 X 3. taken in order from the left. 

7 . r+Yx 2 —rr^3 = 4 . 

8. (25 — 3x5+11 — 4X2) — 4. 

13. 4 X (15 X 7 = 35) + 17-2 X 18 -e- 5. 

14. 10 X 9 15 — (4+~8 — 7 — 2) X 2. 


70. REVIEW OF FUNDAMENTAL OPERATIONS. 

1. Name the results of each fundamental operation. 

2. What kind of numbers can be added or subtracted ? 

3 . Make the signs to indicate each operation. 

4. Make the signs for equality, dollars, and aggregation. 














78 


THE 20tli CENTURY ARITHMETIC. 


5. A student studies or recites 6 hours per day during 5 
days of each week for 4 college sessions of 86 weeks each. 
How many hours does he work during the entire course ? 

6. A and B bought 284 acres of land. A’s share was 24 
acres more than B’s. What was the share of each ? 

7. The sum of two numbers is 500. If 125 be subtracted 
from their sum, the result will be 75 more than the larger 
number. What is the smaller number ? 

8. Of what number is 865 both the divisor and quotient ? 

9. Out of 9,642,000 boys in the United States, if cigar¬ 
ettes kill 1 and injure 15 in every 100, how many remain 
uninjured from this cause ? 

10 . A pleasure party, consisting of 12 women and 8 men, 
remained out 14 days. The expenses averaged a dollar a 
day each. The men shared equally the whole bill. What 
was each man’s share ? 

11, A school boy is 10 minutes late every day. How 
many minutes does he lose in a session of 180 days ? At 60 
minutes to the hour and 5 hours to the day, to how many 
school days is this equal ? 

12 When a train is moving 50 miles an hour, how many 
telegraph poles will it pass per minute ? 1 mile = 1760 

yards. Telegraph poles are placed 66 yards apart. 1 hour 
= 60 minutes. 

13 . A man drove to town at the rate of 8 miles per hour 
and after remaining awhile, returned at the rate of 6 miles. 
He was traveling just 6 hours. How far was it to town ? 

14. If 26 be used as an addend 39 times, how much less 
than 1225 will the sum be ? 

15. Simplify (30 2 -f- 3 — 4) X 9 + 8 — 3X6. 

16. A rabbit is 45 yards ahead of a dog, but can run only 
25 yards while the dog runs 28 yards; how many yards must 
the dog run to catch the rabbit ? 



PROPERTIES OF NUMBERS. 


79 


71. Some Properties of Numbers. 


The remainder obtained after dividing a number by 9 is the 
same as that obtained after dividing the sum of its digits by 9. 

417 9 gives a remainder 8 , The sum of 4, 1, and 7 is 

12, and 12 9 also gives a remainder 8. But the remain¬ 

der of 12 -P- 9 is found by adding the digits of 12. Hence 
the remainder of any number divided by 9 may be found by 
adding the digits of the number; if this sum be greater than 
9, add its digits, and so on. 

Number. Sum of digits 

of number of 48 of 12 

7,888,976 48 12 3 

and 8 is the remainder after dividing 7,888,976 by 9. 


72. The reason for the statement in italics is as follows: 
Dividing 10, 100, 1000, etc., by 9 gives always a remainder 1. 

Dividing 20, 200, 2000, etc., by 9 gives always a remainder 2. 

Dividing 80, 300, 8000, etc., by 9 gives always a remainder 3. 

Dividing 40, 400, 4000, etc., by 9 gives always a remainder 4, 

etc. 

[Try some of these divisions.] 

3000 i 9 — 333 9’s + a remainder 3. 

100 -4- 9 = 11 9’s + a remainder 1. 

20 :-f-' 9 2 9’s -f a remainder 2. 

1 -f- 9 = 0 9’s + a remainder 1. 

3121 :• 9 = 346 9’s + a remainder 7. 

But 7 is the sum of the digits of 3121. 


73. What will be the remainder when each of the follow¬ 


ing numbers is divided by 9 ? 


1. 1492. 

2. 5280. 

3. 1776. 

4. 3141. 


5. 24,899. 

6. 31,416. 

7. 43,560. 

8 . 84,263. 


9. 78,631. 

10. 42,590. 

11. 78,133. 

12. 65,482. 


13. 1664. 

14. 1733. 

15. 2541. 

16. 6479. 





80 


THE mh CENTURY ARITHMETIC. 


74. This property of the number 9 gives an excellent 
method for proving the fundamental operations and of 
special practical use as applied to multiplication. It is 
called casting 1 out nines. 


ADDITION. Adc 

1776 = 197 9’s + 8 . 
5280 = 586 9’s + 6 . 
1607 = 178 9’s + 5. 
8142 =• 349 9’s + _1_. 
11805 = 1310 9’s -f 15. 
= 1311 9’s + 6 . 


Directions. —Divide each addend by 
9, writing only the remainders. Add 
the remainders and, if larger than 9, 
divide it by 9. The remainder thus 
obtained should be the same as that 
obtained when the sum to be proved 
is divided by 9. It is unnecessary to 
find or write the numbers 197, 586, etc. 


This method of proof could be employed by using other 
numbers as well as 9; 9 is used because of the ease with which 
remainders may be found when a number is divided by 9; 
and it is the remainders only that are needed in the proof. 

Using the numbers in Art. 73, find sums and prove them 
by casting out nines. 

Use examples in addition and prove the sums by casting 
out nines. 


Subtraction. 

87,162 

61,753 

25,409 


Subtract. 

g 

4 The example will explain itself. 

9 Use examples under subtraction. 


Multiplication. Multiply 
54743 5 

273 3 

164229 15 6 

383201 
109486 


Use examples under mul¬ 
tiplication. 


14944839 


6 






PROPERTIES OF NUMBERS. 


81 


Division. 5 

3 653 

3451225553 
2070 

'1855 

1725 

1303 

1035 


268 


345 X 653 + 268 = 225,553. 
Cast out nines from each number. 

3X5+7 4 

15 + 7 
22 

4=4 


Rule. —Cast out 9’s from divisor, quotient, and remain¬ 
der. Multiply together the numbers thus obtained from the 
divisor and quotient, add the number obtained from the 
remainder, and cast out the 9’s from this sum. The'num¬ 
ber obtained should be the same as that obtained from cast¬ 
ing out the 9’s from the dividend. 

75. Where the result of addition, subtraction, etc., is too 
great or too small by an exact number of 9’s, this method 
evidently fails to prove ; for casting out the 9’s gives still the 
same remainder as if the answer were correct. But thp 
probability of making such an error is so small that great 
confidence may be placed in the method. 

In the following examples tell by the method of casting 
out 9’s whether the answers are correct or not. 


1 . 

1728 

365 

5280 

Supposed 7473 
sum 

6 . 

835 

472 

Supposed 4 ^ 
difference 


2. 

3. 

449 

7284 

876 

9782 

984 

8447 

2319 

25513 

7. 

8. 

4372 

7532 

2547 

4265 

1725 

3267 


4. 

5. 

684 

3845 

365 

309 

492 

1849 

1541 

6103 

9. 

10. 

84,273 

71,543 

56,647 

46,271 

28,626 

25,272 













82 



THE 20th CENTURY ARITHMETIC. 





Supposed 



Supposed Remain 




product. 



quotient. 

der. 

11 . 

8617 

X 

245 886,165. 

16. 

1541 -T- 

23 53 


12. 

528 

X 

28 12,154. 

17. 

9855 -r- 

■27 365 


13. 

9148 

X 

478 4,828,629. 

18. 

53,367-^ 

231 231 

6 

14. 

6418 

X 

291 1,867,688. 

19. 

7200 -r- 

225 32 


15. 

7401 

X 

881 54,086. 

20. 

24,192 -P 

432 44 

11 


Casting out 9’s to prove multiplication is a very practical 
operation and should be much used. It is not of so much 
importance as applied to the other operations. 

76. Diversions. — l. Think of a number. Subtract 
from it the sum of its digits: strike out any digit from the 
remainder; tell me the sum of the remaining digits and I will 
tell you the digit struck out. 

Directions.— Subtract the sum of the digits of the number told you 
from 9: the remainder will be the digit struck out. 

2. Think of any number; divide it by 9 and tell me the 
remainder. Strike out any digit from the number thought 
of, divide by 9, tell me the remainder and I will tell you the 
digit struck out. 

Directions. —If the second remainder is less than the first, the 
digit struck out is the difference of the remainders ; but if the second 
remainder is the greater, add 9 to the first remainder and then take 
their difference. 

3 . Think of a number composed of two digits not the 
same. Transpose the digits: take the difference between 
this number and the one thought of: name one of the digits 
and I will name the other. 

Directions. —Subtract the one he names from 9. 

4. Think of any number. Divide it by 9 and tell me the 
remainder. Multiply the number thought of by (here I 
name a multiplier). Divide the product by 9 and the re¬ 
mainder will be (here I name the remainder). 



PROPERTIES OF NUMBERS. 


Directions. —Multiply the first remainder by the multiplier; di¬ 
vide the product by 9 and the remainder will be the. one required. 

5. Think of a number greater than 3. Multiply it by 3: 
if the product is even divide by 2; if odd, add 1 and then 
divide by 2. Multiply the quotient by 3: if this product is 
even, divide by 2; if odd, add 1 and then divide by 2. Now 
divide by 9, tell me the quotient (without the remainder) 
and I will tell you the number thought of. 

Directions. —If even both times multiply the quotient by 4. If 
odd both times, multiply by 4 and add 3. If even first and odd second 
multiply by 4 and add 2. If odd first and even second, multiply by 4 
and add 1. 


77. 

A number is either : 


( Odd or 1 

) Prime or 1 

) Integral or 

( Concrete or j 

) Cardinal or 

( Even i 

| Composite i 

^ Fractional 

(Abstract i 

\ Ordinal. 


a. An odd number is one not divisible by 2; as 9, 45. 

b. An even number is one divisible by 2; as 8, 76. 

C. A prime number is one divisible only by itself ; as, 7, 29. 

' All prime numbers except 2 are odd. 

Two numbers are prime to each other when no number 
will divide them both. Thus, 8 and 15, though not prime 
numbers, are prime to each other. 

d. An integral number, or an integer , is a whole num¬ 
ber; as 9, 22. 

e. A fractional number is one containing a fraction; as, 
f, 5*, 7.4 


78. Tests of Divisibility. 

A number is divisible— 

By 2 when its last digit is even or naught. 18, 130. 



84 


THE 20th CENTURY ARITHMETIC. 


By 4 when the number expressed by the last two digits is 
divisible by 4. 10812, 1724. 

By 8 when the number expressed by the last three digits is 
divisible by 8. 71824, 18016. 

By 3 when the sum of its digits is divisible by 3. 645 is 

divisible by 3 because 6 + 4 + 5 == 15 is divisible by 3. 

By 9 when the sum of its digits is divisible by 9. 7 + 3 + 
8 _|_ 9 — 27. Therefore any number composed of the digits 
7, 3, 8, and 9 is divisible by 9; as, 8379, 9873. 

By 5 when the last digit is 5 or 0. 885, 710. 

Since a number whose last digit is 0 is divisible by both 2 
and 5 it is also divisible by 2 X 5 = 10. 

By 6 when it is divisible by both 2 and 3; that is, when 
the last digit is naught or even and the sum of the digits is 
divisible by 3. 

By 7 by trial only. 

By 11 when the difference between the sums of the digits 
in the even and odd places is 0 or divisible by 11. 782,056 is 

divisible by 11 because 6 + 0 + 8 = 14, 5 + 2 + 7 = 14, 
and 14 — 14 — 0. 


79. Examples. 


Tell by inspection what numbers will divide 

a. b. c. 

d. 

l. 57. 

927. 

1793. 

1515. 

2. 38. 

279. 

326. 

1818. 

3. 85. 

87'. 

1803. 

5555. 

4. 138. 

604. 

2931. 

16,824. 

5. 428. 

5608. 

1772. 

46,266. 

6. 7288. 

3421. 

8271. 

77,110. 



FACTORS. 


85 


FACTORS. 

80. A factor is one of two or more integers which are 
used to make a product. 

2 X 3 X 5 = 30. Either 2, 3, or 5 is a factor of 30. 

A factor will always exactly divide the product and give 
an integral quotient. 

What are the factors of 15 ? of 21 ? of 6 ? of 14 ? 

a. A number that has factors besides itself and 1 is called 
composite and is a multiple of each factor. 

b. A power is the product when the same factor is re¬ 
peated. 5x5x5 = 125; 125 is a power of 5. 

The factor itself is the first power. When the factor is used twice, 
the product is the second power. When the factor is used three 
times, the product is the third power. When the factor is used four 
times, the product is the fourth power, etc. 

C. Instead of 5 X 5, we may write 5 2 ; 5 to the second 
power. Instead of 5X5X5, we may write 5 3 ; 5 to the 
third power. Instead of 5 X 5 X 5 X 5, we may write 5 4 ; 
5 to the fourth power, etc. 

d. The small figure placed above and to the right to show 
how many times the factor is to be repeated is called an 

exponent. 


81. Prime Factors of a Number. 

To resolve a number into its prime factors is to find what 
prime numbers multiplied together produce it. 



86 


THE goth CENTURY ARITHMETIC. 


What are the prime factors of 8080 ? 


2|8080 
2 11540 
2| 770 
51 885 
71/77 
11 


Directions. —Divide the given number by any prime 
number which, by inspection, you know will divide it. 
Continue thus to divide the successive quotients till the 
the quotient itself is a prime number. The divisors and 
last quotient will be the prime factors sought, because 
they will all be prime numbers and when multipled to¬ 
gether will produce the given number. 


3080 = 2X2X2X 5X7X 11, 

= 2 3 X 5 X 7 X 11. 

/ N. 

The rules in Art. 78 are given to assist in telling when 
a number is divisible by 2, 8, 5, and 11. There is no prime 
number larger than 11 in the following examples: 


What are the prime factors of 


l. 42. 

7. 462. 

13. 726. 

19. 1617. 

2. 80. 

8. 1925. 

14. 9075. 

20. 1815. 

3. 105. 

9. 1575. 

15. 5929. 

21. 8680. 

4. 88. 

10. 2156. 

16. 18,475. 

22. 1980. 

5. 180. 

11. 600. 

17. 11,025. 

23. 1260. 

6. 1155. 

12. 198. 

18. 2750. 

24. 7850. 

For other 

examples employ 

the numbers used 

as terms of the 


fractions in Art. 123. 


82. If any prime number larger than 11 is a factor it 
must be found by trial. The number of trials, however, 
may be limited by this rule: Divide by prime numbers in 
their order of increase till you find a factor, or till the quo¬ 
tient is less than the divisor. In the latter case the number 
itself is prime. For example, has the number 481 factors or 
is it prime ? Dividing 481 in succession by 2, 8, 5, 18, 
17, and 19, we find none of them to be factors and in every case 




FACTORS. 


87 


the quotient is greater than the divisor. [Perform these 
divisions.] Dividing by 23, the quotient is 18 with a remain¬ 
der; hence 431 is a prime number. For, if it were composite, 
both divisor and quotient would be factors. But by trial 
we found that it had no factors less than 23 and since the 
quotients become less as the divisors become greater than 23, 
there is no use to try further. 

83. The following list contains the first thirty prime numbers. 


1 

13 

37 

61 

89 

2 

17 

41 

67 

97 

3 

19 

43 

71 

101 

5 

23 

47 

73 

103 

7 

29 

53 

79 

107 

11 

31 

59 

83 

109 


84. Are the following numbers prime or composite f 


1. 323. 

2. 853. 

3. 756. 


4. 503. 

5. 881. 

6. 997. 


7. 1067. 

8. 601. 
9. 3267. 


10. 463. 

11. 283. 

12. 209. 


Find the prime factors of the following numbers: 

25. 15,925 


13. 11,011. 

14. 9350. 

15. 13,981. 

16. £82. 


17. 882. 

18. 1067. 

19. 1178. 

20. 841. 


21 . 870. 

22. 1843. 

23. 1921. 

24. 8085. 


26. 9350. 

27. 15,675 

28. 1115. 


This process of resolving a number into its prime factors 
is used in finding, by one method, the greatest common 
divisor and least common multiple of numbers; to be ex¬ 
plained later. 


THE goth CENTURY ARITHMETIC. 


Greatest Common Divisor. 

85. A common divisor of two or more numbers is an ex¬ 
act divisor of each number. 

3 is a common divisor of 18 and 27 because it will exactly divide 
each number. 

The Greatest Common Divisor (G. C. D.) of two or more 
numbers is just what the words imply—their greatest com¬ 
mon divisor. 

9 is the greatest common divisor of 18 and 27 because it is the 
greatest number that will exactly divide them both. 

86 . 'We know by inspection that 8 • 5 divided by Ogives 
5 and divided by 5 gives 8. 

A dot placed between numbers means that they are to be multi¬ 
plied together. 5-3 = 5 X 3. 

It is evident, also, that 2 • 8 - 5 = 6 • 5 = 10- 8 = 
2 • 15 = 80. Hence, 30 is divisible by 2, 3, 5, 6, 10 and 15. 

A number is divisible by each of its prime factors and the pro¬ 
duct of any two or more of them and by nothing else. [Commit 
this to memory.] 

70 = 2 • 5 • 7 is divisible by 2, 5, 7, 10, 14 and 35. 

Every number is divisible by itself. 


What numbers will exactly divide 


1. 

2 • 

3 • 

5. 

4. 5 • 

' 7 • 

7. 

7. 2 • 

3 • 

5 • 

7. 

2. 

3 • 

7 • 

11. 

5. 3 • 

5 • 

7. 

8. 3 • 

5 • 

7 • 

11 

3. 

2 • 

5 • 

7. 

6. 3 • 

5 • 

5. 

9. 2 ■ 

3 • 

3 • 

5. 


105 = 3 • 5 • 7. 
165 = 3 • 5 • 11. 



FACTORS. 


89 


105 is divisible by 3, 5, 7, 15, 21, and 35. 

165 is divisible by 3, 5, 11, 15, 33, and 55. 

3, 5, and 15 are the only common divisors, therefore, 15 
is the greatest common divisor. 

87 . The G. C. D. of two or more numbers is the product of 
all the prime factors common to the numbers. 


30 = 

= 2 • 

3 • 5. 


3 and 5 will divide 

them 

all ; hence 

105 = 


3 • 5 • 

7. 

3 X 5 is 

their G. C. 

D. 



330 - 

r 2 ■ 

3 • 5 • 

11. 







What is the G. C. D. of 







5 • 

7. 

2 

• 3 

• 5. 

2 

• 3 

• 5 • 

7. 


l. 5 • 

11. 

4. 2 • 

3 

• 7. 

7. 2 

• 3 

• 5 • 

11. 


5 • 

13. 

2 

• 3 

• 11. 

2 

• 3 

• 5 • 

5. 




• 



2 

• 3 

• 5 • 

3. 


3 • 

7. 

2 

■ 5 

• 7. 

5 

• 7 

• 11 

• 13. 


2. 7 • 

11. 

5. 3 • 

5 

• 7. 

8. 2 

• 5 

• 7 • 

11. 


5 • 

7. 

5 

■ 5 

• 7. 

3 

• 5 

• 7 

• 11. 







5 

• 7 

• 7 • 

11. 


11 • 

18. 

5 

• 7 

• 7. 

2 

• 2 

• 3 

• 3 • 

5. 

3. 2 • 

11. 

6. 7 

• 7 

• 7. 

9. 2 

• 2 

• 3 

• 3. 


11 • 

11. 

7 

• 7 

• 11. 

2 

• 2 

• 3 

• 3 • 

13. 


• 




2 

• 2 

• 2 

• 3 • 

3. 

See Art. 8C 

i, d. 









3 • 

7 2 • 11. 



2 3 • 

5 2 

• 7. 



10. 

7 2 • 

11 • 13 


n. 

2 3 • 

5 2 

• 11. 




5 • 

7 2 • 11. 



2 3 • 

5 2 

• 13. 




90 


THE 20th CENTURY ARITHMETIC. 


2 2 

• 5 2 . 



2 3 • 

3 2 • 

5 • 

7. 

2 5 

• 5 3 . 



16. 2 3 • 

3 2 • 

7. 


2 3 

2 • 

• 7 3 . 
7 2 . 



2 3 • 

2 3 • 

3 2 • 
3 2 • 

13. 
5 • 

5. 

2 3 

• 5 3 • 

7. 


5 3 • 

72 . 

11 . 


2 3 

• 5 2 • 

7. 


17. 5 2 • 

7 3 • 

13. 


2 3 

. 52 . 

7. 


5 4 • 

7 2 . 







5 2 • 

7 4 . 



8 2 

• 5 • 

7. 


3 2 • 

5 3 • 

II 4 . 


8 2 

• 5 • 

7 • 

11 . 

18. 2 3 • 

3 2 • 

5 3 • 

II 2 

8 2 

• 5 • 

7 . 

7. 

2 4 • 

5 2 • 

7 • 

11 . 

3 2 

• 5 • 

18. 


3 3 • 

5 3 • 

7 2 • 

11 . 



2 2 

• 3 2 

. 5 s . 7 3 

• 11 . 




19. 

2 2 

• 3 2 

• 5 3 • 7 3 . 

* 





3 2 

• 5 2 • 

. 73 . ns 






2 2 

• 3 2 

' 5 3 • 7 3 

• 13. 




88. Second Method for Finding G. C. D. 

When it is not easy to resolve the numbers into their prime factors, 
this method is preferable. 

14 = 7 twos. 

g = 4 twos. Since 14 and 8 are both divisible by 2 , 

— 7 T - their difference, 6 , is also divisible by 2 . 

o = 8 twos. 

A common divisor of two numbers is also a divisor of their dif¬ 
ference. 

Since a common divisor of two numbers must divide their 
difference, no common divisor can be greater than their dif¬ 
ference. The G. C. D. of 15 and 18 is their difference, 8. 




GREATEST COMMON DIVISOR. 


91 


What is the G. C. D. of 


1. 

15 

and 

18. 

6. 

48 

and 

56. 

ll. 

63 

and 

72. 

2. 

12 

and 

16. 

7. 

45 

and 

54. 

12. 

49 

and 

56. 

3. 

15 

and 

20. 

8. 

33 

and 

44. 

13. 

63 

and 

70. 

4. 

21 

and 

28. 

9. 

26 

and 

39. 

14. 

84 

and 

91. 

5. 

56 

and 

63. 

10. 

48 

and 

60. 

15. 

88 

and 

96. 

00 

CO 

What 

is the 

Gr. C. 

D. 

of 21 and 35 ? 





Their difference is 14, but 14 will not divide either num¬ 
ber. Now, whatever will divide both 21 and 35 will also divide 
14; it will, therefore, divide the difference between 14 and 
either of these numbers. Take the difference between 14 and 
the smaller number because it will be the smaller difference 
and still be as large as the greatest common divisor could be. 

21—14 = 7, and we find by trial that 7 will divide both 
21 and 35. 

Find the Gr. C. D. of 45 and 63. 

63 Subtract 45 from 63, leaving 18. Try 18. Subtract 
45 18 from 45, leaving 27 ; subtract 18 from 27, leaving 
jg 9. Try 9. It is the G. 0. D. When the remainder 

is larger than the subtrahend, continue to subtract 
the subtrahend. 

9 G. 0. D. 


What is the G. C. D. of 


1 . 

21 and 35 ? 

6. 

45 and 75 ? 

11 . 

63 and 81 ? 

2. 

24 and 40 ? ( 

7. 

45 and 81 ? 

12. 

63 and 91 ? 

3. 

27 and 45 ? 

8. 

45 and 63 ? 

13. 

104 and 136? 

4. 

33 and 55 ? 

9. 

35 and 49 ? 

14. 

169 and 221 ? 

5. 

39 and 65 ? 

10. 

28 and 36 ? 

15. 

49 and 119 ? 


45 

18 

27 



92 


THE mh CENTURY ARITHMETIC. 


90- Division is a short method of subtraction; therefore, 
when many subtractions are to be performed, it is better to 
do them by division. 

Directions. —Divide the larger number by the smaller , then the 
divisor by the remainder continously , until there is no remainder. 
The last divisor is the G. C. D. 


What is the G. C. D. of 133 and 190. 


1 

1331190 
133 2 

57| 133 
114 g 
G. 0. D. 19 |W 
57 


Copy this work and hote whence 
the numbers come. The arrange¬ 
ment on the right is to be pre¬ 
ferred. The quotients are of no 
consequence when the work is fin¬ 
ished. 


133 

190 

114 

133 

19 

57 

G. C. D. 

57 


91, Students will have no difficulty in finding the G. 0. D. They 
may omit this explanation and learn it when more advanced. 

Explanation. —Since 9 is a divisor of 45 =5 nines, 

and of 27 — 3 nines, 
it is also a divisor of 45 + 27 = 8 nines, 
and of 45 — 27 = 2 nines. 

Principle I.— A divisor of two numbers is also a divisor of 
their sum and of their difference. 

9 will divide 18, it will divide 18 + 18 (Prin. I.), or any 
number of eighteens. Hence, 

Principle II.— A divisor of any number is also a divisor of 
any multiple (Art. 95.) of that number. 

Now, apply these principles to the example above. 

The G. C. D. of 133 and 190, whatever it is, will divide 
their difference (I.). It will also divide 2 times 57 (II.). 







GREATEST COMMON DIVISOR. 


93 


Since it will divide 188 and 114, it will divide their differ¬ 
ence, 19 (I.). Hence, it cannot be greater than 19. But 
19 will divide 57; therefore, it will divide 2 times 57 (II.), 
and 114 + 19 = 188 (I.), and also 138 + 57 = 190 (I.). 

Hence, since the G. C.D. of 133 and 190 cannot be greater 
than 19 and 19 will divide them, it is their G. C. D. 


92. Geometrical Explanation. 


The problem is, find the longest distance which, when ap¬ 
plied to A and B, will be contained exactly in each of them; 
that is, find the G. C. D. of A and B. Now, whatever will 
measure A and B will measure that part of B which is just 
as long as A, and, consequently, the remainder of B. But 
whatever will measure the remainder, R, cannot be greater 
than R; hence the G. C. D. of A and B cannot be greater 
than R. But will R divide them? If it will divide the 
smaller line A, it will also divide the larger line B; for, if it 
divides A, it will divide that part of B which is just as long 
as A and, of course, itself; that is, A + R, which — B. 

Applying R to A, it is contained in it twice. Therefore, 
R is also contained in B and is the G. C. D. of A and B. 

Examples. 

What is the G. C. D. of 

1. 187 and 323? 3. 700 and 756? 5. 582 and 1067? 

2. 133 and 209? 4. 882 and 1575? 6. 841 and 870? 





94 


THE mh CENTURY ARITHMETIC. 


7 . 1078 and 4181? 10. 1848 and 1921? 13. 990 and 8267? 

8. 118 and 791? n. 823 and 1615? 14. 8085 and 15,925? 

9. 890 and 1869? 12. 420 and 1115? 15. 9350 and 15,675? 

For other examples see Art. 100. 

93 . To find the G. C. D. of several numbers, find the 
G. C. D. of any two of them, then of that result and a third, 
and so on till all the numbers have been used. The last 
G. C. D. will be the one sought. For, each G. C. D. will 
contain all the factors common to the numbers that have 
been used, and the last G. C. D. will contain all the factors 
common to all the numbers. 

What is the G. C. D. of 285, 190, and 133 ? 

The G. C. D. of 285 and 190 is 95; of 95 and 133 is 19; 
therefore, the G. C. D. of 285, 190, and 133 is 19. 

Examples. 


What is the G. C. D. of 

1. 4977, 3555, and 6636? 

2. 1827, 3906, and 4599 ? 

3. 1273, 6231, and 4087? 

4. 791, 1243, and 1469? 

5. 3850, 7700, and 10,780? 


6. 6827, 5985, and 5130 ? 

7. 4578, 5986, and 6867 ? 

8. 407, 814, and 444 ? 

9. 915, 1098, and 1830 ? 

10. 730, 438, and 886 ? 


94 . Practical Applications. 

1. What is the greatest equal distance apart that posts 
can be placed around a triangular-shaped field whose sides 
are 840, 408, and 760 feet ? Draw a triangle. 

2. A man owns three farms containing respectively 270, 
594, and 945 acres. He wishes to divide them into equal 



LEAST COMMON MULTIPLE. 


95 


sized lots, each lot to contain as many acres as possible. 
How many acres will each lot contain ? 

3. A courtyard which measures 1197 inches on two sides 
and 1468 inches on the other two, is to be paved with square 
tiles as large as possible. What will be the size of each tile? 

4. What is the greatest number that will divide 228 and 
244 and give the remainder 13 from each ? 

5. A man has on deposit $625, $875, and $1125 in three 
different banks. What is the largest sum he can draw, if 
he draws on his deposits for equal amounts each time until 
the deposits are exhausted ? 

The G. C. D. of numbers is used chiefly in reducing frac¬ 
tions to their lowest terms. See Art. 123. 


95 . Least Common Multiple. 

A multiple is the result obtained by multiplying. 

This is also called a product, but the word multiple is used with 
reference to one factor only, whereas, a product is used with reference 
to all the factors. 

a. A multiple of a number is exactly divisible by the 
number. 

12 is a multiple of 3 because it is divisible by 3. 

Name some, multiple of 

5, 8, 2, 4, 7, 8, 9, 11, 6, 12. 

f 

Of what number is 6 a multiple ? Of what number is 
9 ? 10 ? 12 ? 15 ? 18 ? 21 ? 24 ? 

Is 17 a multiple of 5 ? Is 24 a multiple of 8 ? Is 57 a 
multiple of 9 ? 



96 


THE 20th CENTURY ARITHMETIC. 


b. A common multiple of two or more numbers is a num¬ 
ber which is divisible by each given number. 24 is a common 
multiple of 3 and 4 because it is divisible by each of them. 

Give some common multiple of 

2 and 8. 4 and 6. 8 and 4. 9 and 12. 

2 and 5. 8 and 6. 5 and 7. 10 and 15. 

C. The Least Common Multiple (L. C. M.) of two or more 
numbers, is the least number divisible by each of them. 

The multiples 

Of 6 are 6, 12, 18, 24, 80, 36 , 42, 48, 54, 6.0, 66, 72, etc. 

Of 9 are 9, 18, 27, 36 , 45, 54, 68, 72, etc. 

Of 12 are 12, 24, 36 , 48, 60, 72, etc. 

36 is the L. C. M. of 6, 9, and 12, because it is the least 
number divisible by each of them. 

The number whose factors are 2, 8, and 5 is divisible by 
each of these factors and by their product taken every way 
possible. Therefore, 2 • 3 • 5, is the L. C. M. of 2, 3, 5, 6, 
10, and 15. 


3 • 

5 • 

7 is the L. C. M. of 15, 

21, and 35. 

Of what is 

l. 8 • 5 the L. C. M. ? 6. 

3 ■ 

• 5 • 

7 the L. C. M. ? 

2. 2 

• 5 

• 7 the L. C. M. ? 7. 

3 • 

5 • 

11 the L. C. M. ? 

3. 5 

• 7 

• 11 the L. C. M. ? 8. 

2 • 

■ 7 • 

11 theL. C. M. ? 

4. 3 

• 7 

• 11 the L. C. M.? 9. 

2 • 

• 3 ■ 

• 7 the L. C. M. ? 

5. 2 

• 5 

• 11 the L. C. M.? io. 

5 

• 7 ■ 

• 11 theL. C. M.? 

96. 

To Find the L. C. M. of Several Numbers. 


Method I. Resolve each number into its prime factors and 
select from these all the factors, and no more , necessary to produce 
each number. The product of these factors will be the L. C. M. 
sought. 


LEAST COMMON MULTIPLE. 


97 


What is the L. C. M. of 42, 66 , and 105 ? 


42 = 2 • 3 • 7. 

66 = 2 • 3 • 11. 

105 = 3-5-7. 

A. 2-3-7. 

B. 2-3-7-11. 

L. C. M. 2 • 3 • 7 • 11 • 5. 


The factors necessary to pro¬ 
duce 42 are 2,3, and 7 ; set these 
aside, A. To produce 66 we 
must have, besides factors al¬ 
ready in A, the factor 11; and 
to produce 105, the additional 
factor 5. Therefore, 

2 • 3 • 7 • 11 ■ 5 is divisible 
by 42, 66, and 105. 


Art. (repeated) 86 . A number is divisible by each of its 
prime factors and the product of any two or more of them and by 
nothing else. 

A L. C. M. must, therefore, contain all the prime fac¬ 
tors (and no others) found in each of the given numbers. 

.The L. C. M. of 2 • 3 and 2 • 7 must be divisible by 
2 , 3, and 7 and, therefore, must contain these factors. 

97. What is the L. C. M. of [Give factors only of answer.] 



3 • 

5. 

2 • 

3 • 

11 . 


2 3 

3 • 

7. 

1. 

3 • 

7. 

6. 3 • 

5 • 

7. 

n. 

2 3 

5 • 

11 . 


5 • 

7. 

5 • 

7 • 

11 . 


2 3 

5 • 

7. 


2 • 

13. 

2 • 

2 • 

13. 


2 3 

• 3 2 

7. 

2. 

2 • 

5. 

7. 2 • 

3 • 

7. 

12. 

3 2 

5 2 • 

11 . 


3 • 

5. 

3 • 

7 • 

11 . 


5 2 

• 7 • 

11 . 


7 • 

7. 

3 • 

3 • 

5. 


5 2 

74 . 

11 . 

3. 

5 • 

5. 

8. 5 • 

5 • 

7. 

13. 

5 3 

II 3 

• 13. 


5 • 

7. 

3 • 

5 • 

7. 


11 

• 13 4 

• 17. 


2 • 

5. 

3 • 

5 • 

5. 


2 3 

5 2 • 

11 . 

4. 

3 • 

5. 

9. 7 • 

7 • 

11 . 

14. 

3 2 

• 7 • 

11 . 


5 • 

5. 

5 • 

5 • 

7. 


2 3 

• 5 • 

7. 


3 • 

5. 

3 • 

13 

7. 


5 2 

• 7 • 

11 . 

5. 

2 • 

7. 

10. 5 • 

7 • 

13. 

15. 

73 

• ll 2 

• 13. 


5 • 

7. 

5 • 

5 • 

7. 


5 • 

73 . 

13 2 . 





98 


THE goth CENTURY ARITHMETIC. 


98. By inspecting the example solved in Art. 96, it may 
be seen that a factor found in two or more of the given num¬ 
bers is used but once in the L. C. M. For instance, 2 is 
found in both 42 and 66, but is used only once in the L. C. 
M. From this fact we derive 

Method II. Place the numbers in a horizontal row and divide 
by any prime number that will divide two or more. Bring down 
in the row of quotients those numbers that cannot be divided. 
Continue this process till no two numbers have a common factor. 
The product of all the divisors and the last quotients is the L. 
C. M. of the given numbers. 



2 | 

42 

66 


105 



3| 

21 

33 


105 



7| 

7 

11 


35 





11 


5 



L. C. M. = 

2 X 

3 X 

7 X 11 

X 5. 

99 

1 . What is the L. C. 

M. of 




l. 

80, 105,. and 175 ? 


ll. 

16, 

40, 

96, and 105 ? 

2. 

78, 105, and 110 ? 


12. 

M 

5,8, 

4, 5, 6 , 7, 8 , and 9? 

3. 

66 , 105, and 885 ? 


13. 

10 , 

12 , 

15, and 75 ? 

4. 

156, 830, and 385 ? 


14. 

45, 

75, 

135, and 180 ? 

5. 

300, 630, and 735 ? 


15. 

16, 

60, 

140, and 192 ? 

6. 

4, 10, 15, and 25 ? 


16. 

48, 

54, 

112, and 312 ? 

7. 

11, 35, 49, and 77? 


17. 

36, 

42, 

48, and 126 ? 

8. 

1800, 2700, and 4500 ? 

18. 

63, 

72, 

84, and 105 ? 

9. 

15, 34, 102, and 13i 

r ? 

19. 

14, 

35, 

45, and 140 ? 

10 

273, 330, 495, and 546 ? 

20. 

39, 

77, 

231, and 429 ? 





LEAST COMMON MULTIPLE. 


99 


100. Application . 

1. .What is the width of the narrowest walk that can be 
paved with blocks 45 inches long and 30 inches wide, the 
blocks being placed either across or with the length of the 
walk. 

2 . What is the shortest piece of calico that can be cut 
into pieces 6 , 10, or 15 yards long and nothing remain ? 

3. A, B and C ride at a constant rate on a circular bi¬ 
cycle track. A rides round once in 15 minutes, B in 18 
minutes, and C in 20 minutes. If they start together, how 
long before they will be together again ? How many rounds 
will each have made ? 

4 . How many acres in the smallest farm that can be ex¬ 
actly divided into lots of 25, 30, 35, or 40 acres each ? 

5 . The front wheel of a bicycle is 9 feet and the rear 
wheel 6 feet in circumference. A marked spoke in each 
wheel is vertical at the same instant. How far will the bi¬ 
cycle go before the spokes are again both vertical ? 

6 . How far will the bicycle have gone if the spokes have 
both been vertical 293 times ? (See Example 5.) 

101 . Diversions.— l. A wishes to expend the smallest 
equal sums possible in the purchase of sheep at $ 12 , cows at 
$20, and horses at $75 or $90 a head. Which priced horses 
should he buy ? What sum will he expend, and how many 
of each will he buy ? 

2 . An old woman, on being asked how many eggs she had, 
could only remember that in counting them into the basket 
by 2’s, 3’s, 4’s, or 5’s there always remained one over. What 
is the smallest number she could have had ? 


100 


THE mh CENTURY ARITHMETIC. 


102 . REVIEW. 

1. 1894 X 865 — 691,810. Prove by casting out 9’s. 

2. When does casting out 9’s fail to prove ? 

3 . What is a prime number ? An integer ? 

4 . Give the tests of divisibility for 2 , 3, 4, 5, 6 , 9, and 11. 

5 . What is a factor ? A power ? An exponent ? 

0 . Resolve 13,475 into its prime factors. 

7 . Is the number 18,473 prime or composite? 

8 . By what numbers is a given number divisible? Art. 64. 

9 . Find by second method the G. C. D. of 990and3267. 

10 . What is a multiple ? 

11 . Give one method for finding the L. C. M. of several 
numbers. 

12 . Find the L. C. M. of 1729 and 2093. Give reason for 
process. 

13 . How do you find the G. C. D. of three or more num¬ 
bers ? Art. 93. The L. C. M ? Art. 101. 

14 . Compare the product- of 1540 and 8003 with the pro¬ 
duct of their L. C. M. and G. C. D. 

15 . Take two numbers: 11 X 7 and 11 X 13. 

Their G. C. D. is 11. Their L. C. M. is 11 X 7 X 13. 

The product of the numbers is 11 X 7 • 11 X 13. 

The product of their G. C. D. and L. C. M. is 11 • 11 X 

7 X 13. 

It will be noticed that the two products contain the same 
factors and, from the nature of the work, that the product 
of any two numbers is equal to the product of their G. C. D. 
and L. C. M. 

The product of two numbers is 13 2 X 5 X 11- Their L. 
C. M. is 13 X 5 X 11; what is their G. C. D. ? One of the 

numbers is 13 X 5, what is the other ? 

16. The product of the G. C. D. and L. C. M. of two num¬ 
bers is 2695: one of the numbers is 35, what is the other ? 
Find their G. C. D. and L. C. M. 

17 . The L. C. M. of two numbers is 6930; their G. C. D. 
is 45: one of the two numbers is 630, what'is the other? 



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